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WEIGHT,  MEASURE  AND  COINS, 


PROPOSED   TO   BE   CALLED   THE 


TONAL     SYSTEM, 


WITH  SIXTEEN  TO  THE  BASE. 


BY 


JOHN    W.\NYSTR0M,    C.   E 


H 


PHILADELPHIA: 

B.    LIPPINCOTT    &    CO, 
LONDON:  TRUBNER  &  CO. 
1862. 


\ 


Entered  according  to  Act  of  Congress,  in  the  year  1862,  by 

JOHN    W.    NYSTROM, 

In   the   Clerk's   Office  of  the   District   Court   for  the   Eastern   District  of 

Pennsylvania. 


KING    &    BAIRP,    PRINTEKS,    SANSOM    ST.,    PHILADELPHIA. 


To  the  International  Association 

for  obtaining  a  Uniform  Decimal  System 

of  Weights,  Measures  and  Coins : 
Gentlemen: — 

The  substance  of  this  book  was  laid  before  the  meet- 
ing held  by  the  British  branch  of  your  Association,  at 
Bradford,  England,  on  the  11th  of  October,  1859.  I 
have  since  made  great  effort  both  in  England  and 
America  to  have  the  same  published,  but  have  not 
succeeded  until  now,  when  done  at  my  own  expense. 
I  incur  constant  outlays  for  the  general  interest  in  pro- 
moting science  and  the  arts,  for  which  I  expect  no 
remuneration. 

Scientific  men  of  good  standing  have  remarked  that 
it  would  not  be  proper  to  publish  the  correspondence 
with    the   Decimal   Association,    and   that    with    the 

Society  of  Philadelphia,  but  I  am  of  a  different 

opinion.  I  do  not  publish  the  correspondence  referred 
to,  for  the  sake  of  showing  which  of  us  are  right  or 
wrong,  but  simply  for  the  discussion,  which  I  consider 
to  be  of  some  importance  in  digesting  the  subject, 
and  it  may  save  others  from  making  similar  remarks 
on  the  same.  The  remarks  made  on  my  new  system 
of  arithmetic,  by  the    Decimal   Association,  and    the 

Society  are  of  very  little  importance  further  than 

of  giving  rise  to  discussion  on  the  subject,  but  they 

M 510946 


have  proved  of  great  importance  in  preventing  the  pub- 
lication of  the  herein  called  tonal  system. 

I  shall  ahvays  be  very  glad  to  hear  any  remarks  on 
the  subject,  as  I  am  well  convinced  that  the  more  it  is 
attempted  to  defend  the  decimal  arithmetic,  the  more 
its  folly  will  be  exposed. 

When  the  book  is  published,  one  copy  shall  be  sent 
to  each  member  of  the  International  Decimal  Associa- 
tion, and  I  shall  feel  gratified  and  compensated  if  it 
receives  your  attention. 

The  noble  object  of  your  Association  deserves  all 
possible  success,  and  it  is  your  agreeable  duty  to 
endeavour  to  relieve  us  of  the  present  dreary  and 
complicated  calculations,  and  to  establish  such  system, 
as  in  all  its  bearings  would  become  the  most  simple 
and  efficient  for  all  classes  of  mankind,  and  our 
descendants  will  thank  you  for  ever. 

If  I  knew  of  no  better  than  the  decimal  arithmetic, 

I  would  get  along  very  well  with  it,  but  now  unfortu 

nately,  when  I  get  entangled  in  the  complication  with 

long  decimal  tails,  I  grumble  long  verses  over  it. 

I  have  the  honor  to  subscribe  myself. 

Your  most  obedient  servant, 

John  W.  Nystrom, 

Engineer. 
Philadelphia,  in  January,  1862. 


^rtsibtnts  of  Ijjc  |ntemittonal  §tt\\m\  Association. 

The  BARON  JAMES.'^DE  ROTHSCHILD,  Consul-General  of  the  Austrian  Empire. 

Witt  ^rcsibfuts. 

For  Belgium...COV^T  ARRIVABENE,  Brussels. 

M.  DE  BROUCKERE,  Burgomaster  of  Brussels. 

J.  LIA6RE,  Major  of  Engineers. 

A.  QUETELET,  Astronomer  Royal., 

J.  S.  STAS,  Professor  of  Chemistry  in  the  i^cole  Militaire,  Brussels. 

France ELIE  DE  BEAUMONT,  Senator,  Member  of  the  Institute. 

5IIGHEL  CHEVALIER,  Councilor  of  State,  Member  of  the  Institute. 
M.  LE  PLAY,  Councilor  of  State. 
C.  L.  MATHIEU,  Member  of  the  Institute. 
GENERAL  MORIN,  Member  of  the  Institute. 
EMILE  PEREIRE,  Paris. 
German  Zollverein.    DR.  STEINBEISS,  Privy  Councilor,  Stuttgart. 
Great  Britain..niS  GRACE  RICHARD  WHATELY,  D.  D.,  Archbishop  of  Dublin. 
RIGHT  HON.  THE  EARL  OF  ROSSE,  K.  P.,  F.  R.  S. 
RIGHT  HON.  BARON  FORTESCUE. 

THE  VERY  REV.  RICHARD  DAWES,  M.A.,  Dean  of  Hereford. 
RICHARD  COBDEN,  Esq..  M.P. 
JAMES  YATES,  Esq  ,  M.  A.,  F.  R.  S. 

Greece 

Holland M.  VROLIK,  formerly  Minister  of  Finance,  The  Hague. 

Ital)/ HIS  EXCELLENCY  THE  MARQUIS  D'AZEGLIO,  London. 

CHEVALIER  BERTINI,  Turin. 

SIGNOR  BARTOLOMEO  CINI,  Florence. 

CHEVALIER  CORRIDI,  Director  of  the  Technical  Institute,  Florence. 

SIGNOR  ENRICO  MAYER,  Pisa. 

DR.  PANTALEONE,  Rome. 

Liberia J.  J.  ROBERTS,  Esq.,  late  President  of  the  Republic,  MonroTia. 

Mexico SIGNOR  PACHECO,  Minister  Plenipotentiary  at  Paris. 

Portugal 

M.  D'AVILA,  Minister  of  State,  formerly  Minister  of  Finance. 
JOAQUIM   HENRIQUEZ  FRADESSO   DA    SILVEIBA,   Superintendent  of 
Weights  and  Measures,  Lisbon. 

Russia MR.  KUPFFER,  Member  of  the  Imperial  Academy  of  Science,  St.  Peterfburg. 

Spain SIGNOR  RAMON  DE  LA  SAGRA,  Member  of  the  Cortes. 

Switzerland.. ..VROVESSOK  DE  LA  RIVE,  Geneva. 
GENERAL  DUFOUR,  Geneva. 
M.  TRUMPLER,  Zurich. 
United  States  of  North  America. 

DR.  A.  D.  BACHE,  Washington. 
HON.  GEORGE  BANCROFT,  New  York. 
HICKSON  FIELD,  Esq.,  New  York. 
HON.  CHARLES  SUMNER,  Boston. 

Secretaries. 

For  Enffland...B.E'SKY  COLE,  ESQ.,  South  Kensington  Museum. 

France H.  HIPPOLYTE  PEUT,  12  Rue  de  la  Bruyere,  Paris. 

miited  Sfates...T'l.  A.  VATTEMARE,  17  Rue  de  Tivoli,  Pa 


CONTENTS. 


-♦•♦- 


PAGE. 

Letter  to  the  International  Decimal  Association 3 

Presidents  of  the  International  Decimal  Association 5 

Introduction 9 

Tonal  System  of  Arithmetic 15 

Names  of  the  Tonal  Figures 16 

Tonal  and  Decimal  Numbers  compared 18 

Tonal  and  Decimal  Fractions 21 

Tonal  Addition  and  Multiplication  Tables 22 

Addition 24 

Subtraction  and  Multiplication 25 

Division 26 

Tonal  Logarithms 27 

Tonal  Weight,  Measure,  and  Coin 27 

Circumference  of  the  Earth 29 

Length,  Tonal  Meter 30 

Time  and  the  Circle 32 

Tonal  Watch,  or  Clock-dial , . . . .  34 

Tonal  Compass 35 

Latitude  and  Longitude,  Tonal  Division 36 

Measure  of  Surface 36 

"             Capacity 37 

"            Weight 37 

"            Power 38 

Money 39 

Market  Prices 41 

Postage  Stamps. 42 


8 


Division  of  the  Year 43 

Measure  of  Heat 44 

Music 45 

Boem's  Flute 50 

Abbreviation  of  Tonal  Units 51 

Examples  for  Tonal  Calculation 52 

Counting  Machine  for  the  Tonal  System 57 

Russian  Tschoty 59 

Meeting  of  the  International  Decimal  Association  in  England 61 

Letter  from  the  British  Branch  of  the  International  Association. ..     62 

Letter  to  the  International  Decimal  Association 65 

Critic  on  the  French  Meter 75 

Verniers,  Tonal  and  Decimal 81 

Weights  for  Weighing 82 

Correspondence  with  the Society  of  Philadelphia 85 

Comments  on  the  Tonal  System 87 

Reply  to  the  Comment  on  the  Tonal  System 89 

Binary  Division,  Discussion  on  the  Term 90 

Sixths  and  Thirds  in  the  Tonal,  Decimal  and  Octonal  Systems 91 

Prices  on  Street  Railroads 92 

Railroad  Times 93 

Market  Prices,  Discussion  on 94 

American  Dollar  a  Medium  Unit 95 

Market  Practice,  Discussion  on 97 

Mitchell's  Price  Ticket-box 98 

Mr.  Taylor's  Octonal  System 101 

Tonal,  Decimal,  and  Octonal  Systems 102 

Calculating  Machine,  Nystrom's 105 

Pocket-Book  of  Mechanics  and  Engineering 106 


INTEODUCTION. 


The  introduction  of  the  decimal  system  of  weight, 
measure,  and  coins,  is  steadily  progressing  in  most  parts 
of  the  world,  but  when  or  wherever  it  is  first  proposed, 
it  meets  with  many  natural  and  reasonable  objections. 
The  inconvenience  of  the  decimal  arithmetic  is  well 
known,  and  better  bases  for  the  same  have  been 
frequently  proposed,  but  there  has  not,  that  I  am 
aware  of,  been  made  any  earnest  attempt,  by  proper 
authority,  to  introduce  a  system  that  would  in  all  its 
bearings  constitute  the  greatest  possible  simplicity  and 
efficacy,  nor  to  remove  the  principal  objections  to  the 
decimal  arithmetic.  Questions  may  arise,  first,  what 
are  the  difficulties  and  objections  ?  and  secondly,  how 
can  they  be  removed  and  overcome  % 

The  principal  difficulty  and  objection  to  the  decimal 
system  is,  that  the  base  10  does  not  permit  of  binary 
divisions,  as  required  in  the  shop  and  the  market.  In 
attempting  to  introduce  the  decimal  system  in  England, 
it  met  the  said  reasonable  objections  by  Lord  Over- 
stone's  observation  that  "  the  number  12  presents 
"  greater  advantage  than  10 ;  a  coinage  founded  on 
"  the  first  number  is  more  convenient  for  the  purpose 
"  of  the  shop  and  market."  It  is  evident  that  12  is  a 
better  number  than  10  or  100  as  a  base,  but  it  admits 
of  only  one  more  binary  division  than  10,  and  would, 
therefore,  not   come  up  to   the   general   requirement. 


10 

The  number  16  admU  binary  division  to  an  infinite 
extent,  and  would,  therefore,  be  the  most  suitable 
number  as  a  base  for  arithmetic,  weight,  measure,  and 
coins. 

The  experience  of  practical  men  and  close  observers 
is,  that  wherever  the  decimal  system  is  introduced,  it 
is  of  more  injury  or  inconvenience  to  the  public  than 
of  benefit  to  the  few  who  have  to  do  with  quantities 
merely  by  figures  on  paper. 

It  is  very  difficult  for  the  ordinary  uneducated 
classes  to  understand  the  decimal  system,  because  they 
want  to  divide  their  things  into  the  most  natural  frac- 
tions, halves,  quarters,  eighths,  sixteenths,  &c.,  &c., 
for  which  10  is  not  a  suitable  number. 

For  calculation  on  paper,  the  decimal  system  is  very 
convenient,  when  it  is  not  necessary  to  understand  the 
operation  or  to  impress  the  value  of  quantities  on  the 
mind,  as  is  the  case  with  many  arithmeticians,  who 
majiages  the  figures  and  comes  to  the  result  as  easy  as 
a  musician  who  plays  the  crank-organ.  The  decimal 
system  is  not  so  easy  for  the  practical  man  and  self- 
thinker,  who  impresses  the  value  and  relative  position 
of  quantities  on  his  mind,  as  he  proceeds  in  his 
measurement  and  calculation.  I  have,  from  my  early 
youth,  had  a  great  deal  to  do  with  different  kinds  of 
measurement  and  calculation,  and  have  always  found 
it  inconveniently  arranged,  not  for  the  different  multi- 
plication and  division  of  the  units  of  measures,  but 
principally  on  account  of  the  arithmetical  system  not 
being  well  planned. 

About  twelve  years  ago  I  invented  a  calculating 
machine,  which  was  patented  in  America  in  the  year 
1850;  in  working  out  this  instrument,  a  great  many 


u 

ideas  suggested  themselves  as  to  the  improvement  of 
the  arithmetical  system,  weight  and  measure  in  general. 
In  a  pamphlet  printed  in  Philadelphia  in  1851,  describe 
ing  the  calculating  machine,  mention  is  made  about 
systems  of  arithmetics  with  8  and  16  as  the  base,  with 
suggestions  how  to  form  six  new  figures  for  the  latter. 
At  that  time  I  acquired  a  good  practice  in  a  system 
with  16  as  the  base,  and  intended  to  publish  in  the 
Journal  of  the  Franklin  Institute  an  article  similar  to 
the  contents  of  this,  butjvyas  not  fulfilled. 

The  base  10  has  very  likely  originated  from  the  10 
fingers  on  the  hands,  which  latter  are  even  yet  used 
sometimes  in  counting,  but  10  is  actually  the  worst 
even  number  that  could  be  selected;  8  or  12  would 
have  been  much  better,  but  16  the  very  best. 

The  principal  difiiculties  and  objections  to  the  deci- 
mal system  cannot  possibly  be  overcome  without 
changing  the  arithmetical  base,  for  which  I  herein  will 
propose  a  new  system  of  arithmetic,  weight,  measure, 
and  coins,  with  the  number  16  as  the  base,  which 
system  will  be  hereafter  spoken  of  as  the  Tonal  S^stem^ 
because  its  base  10  (16)  is  proposed  to  be  called  Ton. 

Attempts  are  now  being  made  in  most  parts  of 
the  civilized  world,  and  an  association  formed  for 
the  purpose  of  introducing  an  international  decimal 
system  of  weights,  measures,  and  coins.  It  is  then 
just  the  time  to  attempt  to  introduce  a  better  system 
of  arithmetic,  which  would  combine  and  include  all 
requirements  of  all  the  diff"erent  classes  of  mankind. 
With  the  present  arithmetic,  it  is  utterly  impossible  to 
come  to  a  satisfactory  decision  on  a  uniform  system  of 
weight,  measure,  and  coins. 

The  International  Decimal  Association  is  in  favor 


12 

of  introducing  the  French  metrical  system,  which  is 
the  most  complete  in  existence,  but  has  the  evident 
disadvantages  herein  alluded  to. 

In  the  tonal  system^  what  is  now  generally  understood 
by  decimals,  will  become  the  most  natural  for  all  pos- 
sible requirement  without  exception,  as  well  for  meas- 
urement in  the  shop  and  market  as  for  calculation ; 
and  particularly  so  in  mental  calculation;  being  based 
on  the  binary  multiplication  and  division,  makes  it 
most  clear  to  the  mind.  Such  can  never  be  the  case 
with  the  decimal  arithmetic ;  which  will  always  present 
its  difficulties,  and  for  the  same  reason  the  French 
metrical  system  will  never  be  well  received  in  an  English 
or  American  machine  shop ;  it  is  not  well  suited  for  prac- 
tical people.  I  have  myself  a  great  deal  of  measuring 
to  do,  and  always  prefer  the  binary  multiplication  and 
division.  In  the  machine  shop  and  drawing  room  I 
prefer  the  English  foot  with  inches  divided  into 
eighths,  but  on  the  ship's  floor  in  laying  out  lines  of 
vessels  I  prefer  the  English  foot  divided  into  decimals ; 
a  French  meter  would  of  course  answer  the  same  pur- 
pose for  the  latter.  I  shall  never  use  a  French  meter 
in  the  machine  shop  or  drawing  room,  provided  I  am 
not  obliged  to  do  so  by  law,  and  am  sure  that  the 
majority  of  English  Engineers  would  say  the  same. 

Should  the  French  metrical  system  be  introduced  in 
America  and  England,  the  people  would  of  course  in 
time  become  accustomed  to  it,  and  it  would  always  be 
found  to  work  well  by  pen  and  ink,  but  the  binary 
fractions  expressed  by  decimals  would  still  appear 
curious  to  the  majority  of  the  people. 

It  may  be  remarked  that  in  this  age,  almost  every 
one  has  received   more  or  less   education,  and  conse- 


13 

quently  the  decimal  system  would  not  present  such 
inconvenience  as  herein  stated — very  well — the  binary 
system  is  as  natural  for  the  educated  classes  as  for  the 
uneducated,  for  which  it  would  be  the  most  natural  to 
introduce  an  arithmetic  that  would  in  all  its  forms 
become  the  most  convenient  for  all  parties,  and  no 
special  education  required  for  the  same.  It  is  the 
decimal  hase  which  causes  the  mischiefs  and  discordance 
in  weight,  measure,  and  coins,  while  the  tonal  base  would 
create  a  perfect  harmony. 

The  difficulty  of  introducing  the  tonal  system  is  more 
apparent  than  real.  Introduce  it  first  into  schools,  at 
the  same  time  it  will  be  picked  up  by  one  after  the 
other ;  when  a  little  practice  is  acquired,  they  will  soon 
conceive  its  utility  and  simplicity,  and  encourage 
others  to  follow.  At  the  same  time,  the  sixteen  new 
figures  with  their  new  names  and  multiplication  table 
to  be  published  in  all  almanacs  and  newspapers;  the 
Governments  preparing  the  new  standards  for  weight, 
measure,  and  coins ;  the  watch  and  clock  makers  making 
new  time-pieces;  the  mathematicians  preparing  their 
tables  of  logarithms  and  trigonometrical  lines,  &c.,  »&c. 
The  astronomers  preparing  their  tables  and  almanacs 
for  the  land  and  sea  and  celestial  objects;  the  topogra- 
phers altering  their  maps  to  suit  the  new  division  of 
the  globe ;  the  mathematical  instrument  makers  to  alter 
the  angle-measuring  instruments  and  thermometers,  all 
to  suit  the  tonal  system^  and  it  would  soon  be  complete 
for  introduction.  All  the  different  units,  multiplied 
and  divided  by  the  base  16,  could  be  introduced  and 
employed  with  the  decimal  arithmetic  to  begin  with, 
when  in  a  few  years  the  tonal  arithmetic  would  become 
most  natural  with  its  units. 


14 

A  few  weeks  ago  I  read  in  the  London  Engineer  for 
the  25th  of  March,  1859,  an  article  by  the  International 
Association  for  ohtaining  a  uniform  decimal  system  of 
measure^  tveight^  and  coins;  also  Lord  Overstone's  remarks 
in  the  London  Illustrated  News  ;  this  induced  me  to 
resume  my  old  ideas  of  the  arithmetical  system  with 
16  to  the  base,  being  on  a  trip  on  the  river  Volga  from 
Tv8er  to  Tsaritzen  when,  I  had  plenty  of  time  to  devote 
to  this  important  subject,  I  worked  this  out  with  the 
intention  to  submit  it  to  the  above  mentioned  Associa- 
tion. 

The  tonal  si/stem\\exe\\\  proposed  with  six  new  figures 
will  of  course  appear  strange  at  the  first  glance,  and 
may  be  considered  difficult  to  introduce  to  the  public, 
but  a  little  reflection  will  lead  to  the  conviction  of  its 
simplicity  and  importance. 

At  the  end  I  have  given  a  description  of  an  instru- 
ment by  which  the  tonal  system  can  easily  be  acquired, 
and  the  mind  turned  from  the  decimal  base. 

On  the  river  Don,  Cosack,  in  May,  1859, 

JOHN  W.  NYSTROM, 

Engineer. 


PROJECT 

FOR   A 

NEW  SYSTEM  OF  ARITHMETIC 

WITH     16     TO     THE     BASE, 

TO    BE    CALLED    THE 

TONAL   SYSTEM. 

In  the  Tonal  S^/stem  it  is  proposed  to  add  six  new 
figures  to  the  10  arabic,  thus  : 

1,  2,  3,  4,  5,  6,  7,  8,  6,  9,  V,  V,  8,  Zo,  f,  10, 
making  16  characters  to  form  the  base.  In  order  to 
form  a  clear  conception  of  the  nature  and  utility  of  the 
Tonal  System.,  it  will  be  well  to  enter  into  some  details 
of  calculation  with  examples,  in  connection  with  which 
it  is  necessary  to  give  names  to  the  new  figures,  or 
rather  to  give  new  names  to  the  16  characters,  so  as  to 
clearly  distinguish  it  from  our  present  system. 

A  new  system  of  this  kind  could  not  well  be  intro- 
duced in  one  country  alone,  but  the  whole  world  at 
large  must  agree  on  its  acceptance ;  it  then  becomes 
necessary  in  the  project  of  the  system  to  select  such 
names  of  the  figures  as  to  make  it  well  suited  to  all 
languages,  both  in  spelling  and  sound ;  for  which  the 
following  names  are  given,  without  reference  to  any 
language  or  thing.  I  go  so  far  as  to  say  that  I  would 
object  to  a  professor  of  languages  fixing  the  names  of 
the  figures,  because  he  would  surely  select  them  from 
the  Hebrew,  Greek,  or  Latin,  which,  I  have  no  doubt, 
would  make  it  very  pretty,  such  as  Hecatogramme,  in 
the  French  system.  It  is  desirable  to  have  the  names 
clear  and  simple,  in  expressing  as  well  compound  num- 
bers as  the  difi'erent   units  for  measures.     It  is   not 


16 

necessary  to  employ  more  than  one  syllable  for  each 
object  expressed. 

The  names  of  the  Tonal  fig  ares  are  contained  in  the 
following  four  words,  Andetigo^  Suhgrame^  Nikoliuvy^ 
Lapofyton^  which  should  be  learned  by  heart.  The 
vowel  g  in  these  names  should  be  pronounced  as  in  the 
English  word  cglinder,  i  as  in  2uiU,  e  as  in  tJien^  a  as 
in  all. 


Tom 

%l  Names  of  Single  Figures  and 

Compo 

iind  Numbers 

0. 

Noll. 

17. 

Tonra. 

3f. 

Titonfy. 

1. 

An. 

18. 

Tonme, 

40. 

Goton. 

2. 

De. 

1^. 

Tonni, 

43. 

Gotonti. 

3. 

Ti. 

19. 

Tonko, 

46. 

Gotonby. 

4. 

Go. 

n. 

Tonhu. 

4^. 

Gotonhu. 

5. 

Su. 

119. 

Tonvy. 

50. 

Suton. 

6. 

By. 

18. 

Tonla. 

80. 

Me  ton. 

7. 

Ha. 

1^. 

Tonpo. 

^0. 

Huton. 

8. 

Me. 

IT. 

Tonfy. 

m. 

Vytonme. 

6. 

Ni. 

20. 

Deton. 

m. 

Potonfy. 

9. 

Ko. 

21. 

Detonan. 

T5. 

Fytonni. 

I. 

Hu. 

22. 

Detonde. 

100. 

San. 

f. 

Yy. 

24. 

Detongo. 

101. 

Sanan. 

8. 

La. 

26. 

Detonby. 

102. 

Sande. 

g. 

Po. 

28. 

Detonme. 

106. 

Sanby. 

f. 

Fy. 

2^. 

Detonhu. 

lOS. 

Sanpo, 

10. 

Ton. 

22. 

Detonpo. 

110. 

Santon. 

11. 

Tonan. 

30. 

Titon. 

in. 

Santonhu. 

12. 

Tonde. 

31. 

Titonan. 

120. 

Sandeton. 

13. 

Tonti. 

32. 

Tidonde. 

129. 

Sandetonko 

14. 

Tongo. 

35. 

Titonsu. 

130. 

Santiton. 

15. 

Tonsu. 

39. 

Titonko. 

145. 

Sangotonsu. 

16. 

Tonby. 

31^ 

Titonvy. 

200. 

Desan. 

17 


28T. 

38^. 

700. 

Wf. 

1000. 

2000. 

8W5. 

1,0000. 

10,0610. 

10,0000. 

100,0000. 

1510,0000. 

1,0000,0000. 

1,0000,0000,0000. 

1,0000,0000,0000,0000. 

2,885«,7^0f. 


Desan-metonfv. 
Tisan-latonhu. 
Rasan. 

Husan-vytonfy. 
Mill. 
Demill. 

Memill-husan-vy  tonsil. 
Bong. 

Yybong,  bysanton. 
Tonbong. 
Sanbong. 

Mill-susanton-bong. 
iam. 
Song. 
Tran. 

Detam,  memill  -  lasan  -  suton 
-  liubong,  ramill-posanfy. 


This  arrangement  of  expressing  numbers  is  clear 
and  simple,  but  it  requires  some  practice  before  the 
sound  impresses  the  corresponding  value  on  the  mind, 
for  which  it  is  necessary  to  have  a  clear  conception  of 
the  sound  and  value  of  each  figure.  The  object  of  em- 
ploying different  consonants  to  the  names  of  the  figures 
is  to  render  it  more  difficult  to  alter  a  written  number 
from  one  value  to  another ;  it  will  also  make  the 
expression  clearer.  Although  the  old  figures  in  the 
Tonal  System  bears  the  old  value  (except  9)  one  by  one, 
it  will  not  be  so  in  compound  numbers,  as  will  be  seen 
in  the  following  table  I : 


18 


TABLE     I. 

NoUdio 

n  of  Tonal  and  Dec 

imal  Numhers 

• 

Decimal. 

Ton  at. 

Decimal. 

Tonal. 

Decimal. 

Toual. 

Decimal. 

Tonal. 

1 

1 

33 

21 

65 

41 

97 

61 

2 

2 

34 

22 

QQ 

42 

98 

62 

3 

3 

35 

23 

67 

43 

99 

63 

4 

4 

36 

24 

68 

44 

100 

64 

5 

5 

37 

25 

69 

45 

101 

66 

6 

6 

38 

26 

70 

46 

102 

66 

7 

7 

39 

27 

71 

47 

103 

67 

8 

8 

40 

28 

72 

48 

104 

68 

9 

^ 

41 

25 

73 

45 

105 

65 

10 

9 

42 

29 

74 

49 

106 

69 

11 

% 

43 

2% 

75 

4^ 

107 

6^ 

12 

19 

44 

2'!9 

76 

4(9 

108 

6(9 

13 

8 

45 

28 

77 

-48 

109 

68 

14 

2 

46 

22 

78 

4S 

110 

QZo 

15 

T 

47 

2T 

79 

4f 

111 

6T 

16 

10 

48 

30 

80 

50 

112 

70 

17 

11 

49 

31 

81 

61 

113 

71 

18 

12 

50 

32 

82 

52 

114 

72 

19 

13 

51 

33 

•    83 

53 

115 

73 

20 

14 

52 

34 

84 

54 

116 

74 

21 

15 

53 

35 

85 

55 

117 

75 

22 

16 

54 

36 

86 

56 

118 

76 

23 

17 

55 

37 

87 

57 

119 

77 

24 

18 

56 

38 

88 

58 

120 

78 

25 

15 

57 

35 

89 

65 

121 

75 

26 

19 

58 

39 

90 

69 

122 

79 

27 

1^^ 

59 

3^ 

91 

5^. 

123 

7^. 

28 

1(9 

60 

m 

92 

6(9 

124 

7'(9 

29 

18 

61 

38 

93 

68 

125 

78 

30 

U 

62 

3S 

94 

6S 

126 

72 

31 

If 

63 

3T 

95 

6T 

127 

If 

32 

20 

64 

40 

96 

60 

128 

80 

19 


TABLE     I . 


Notation  of  Toned  and  Decimal  Numlers. 


Decimal. 

Tonal. 

Decimal. 

Tonal. 

Decimal. 

Tonal. 

Decimal. 

Tonal. 

129 

81 

161 

91 

193 

101 

225 

21 

130 

82 

162 

92 

194 

102 

226 

22 

131 

83 

163 

93 

195 

103 

227 

23 

132 

84 

164 

94 

196 

m 

228 

24 

133 

85 

165 

95 

197 

'(05 

229 

25 

134 

86 

166 

96 

198 

m 

230 

26 

135 

87 

167 

97 

199 

107 

231 

27 

136 

88 

168 

98 

200 

t^8 

232 

28 

137 

85 

169 

95 

201 

105 

233 

25 

138 

89 

170 

99 

202 

1^9 

234 

29 

139 

S% 

171 

915 

203 

io^ 

235 

U 

140 

m 

172 

91^ 

204 

W 

236 

2'(9 

141 

88 

173 

98 

205 

m 

237 

28 

142 

82 

174 

9S 

206 

'm 

238 

22 

143 

8T 

175 

9T 

207 

'M 

239 

2T 

144 

Z50 

176 

^;o 

208 

80 

240 

TO 

145 

61 

177 

^,1 

209 

81 

241 

n 

146 

52 

178 

%2 

210 

82 

242 

f2 

147 

53 

179 

%Z 

211 

83 

243 

T3 

148 

54 

180 

%A 

212 

84 

244 

T4 

149 

55 

181 

15 

213 

85 

245 

T5 

150 

56 

182 

^6 

214 

86 

246 

f6 

151 

57 

183 

^:7 

215 

87 

247 

T7 

152 

58 

184 

^^8 

216 

88 

248 

T8 

153 

66 

185 

t6 

217 

85 

249 

T5 

154 

59 

186 

m 

218 

89 

250 

T9 

155 

6% 

187 

u 

219 

8^ 

251 

n 

156 

5'(9 

188 

w  ■ 

220 

819 

252 

w 

157 

58 

189 

m 

221 

88 

253 

f8 

158 

5S 

190 

u 

222 

8S 

254 

T2 

159 

5T 

191 

m 

223 

8T 

255 

n 

160 

90 

192 

m 

224 

W 

256 

100 

20 


TABLE     II 


Notation  of  Tonal  and  Decimal  Numbers. 


Decimal. 

Tonal. 

Decimal. 

Tonal. 

Decimal. 

Tonal. 

100 

64 

100,000 

1,8690 

3,584 

200 

200 

1^8 

200,00(1 

3,0840 

3,840 

TOO 

300 

i2r 

300,000 

4,1820 

4,096 

1000 

400 

IdO 

400,000 

6,1980 

8,192 

2000 

500 

lf4 

500,000 

7,9120 

12,288 

3000 

600 

258 

600,000 

1,2720 

16,384 

4000 

700 

2lf 

700,000 

9,9260 

20,480 

5000 

800 

320 

800,000 

1^3500 

24,576 

6000 

900 

384 

900,000 

8,^190 

28,672 

7000 

1,000 

3S8 

1,000,000 

T,4240 

32,678 

8000 

2,000 

780 

2,000,000 

12,8480 

36,864 

1000 

3,000 

^.^-8 

3,000,000 

28,196100 

40,960 

9000 

4,000 

T90 

4,000,000 

38,0100 

45,056 

^000 

5,000 

1388 

256 

100 

49,152 

19000 

6,000 

1770 

512 

200 

52,348 

8000 

7,000 

1^.58 

768 

300 

57,344 

2000 

8,000 

2040 

1,024 

400 

61,440 

fOOO 

9,000 

23(98 

1,280 

500 

65,536 

1,0000 

10,000 

2710 

1,530 

600 

262,144 

4,0000 

20,000 

4K20 

1,792 

700 

524,288 

8,0000 

30,000 

7550 

2,048 

800 

786,432 

19,0000 

40,000 

^840 

2,304 

100 

1,048,576 

T,0000 

50,000 

19-150 

2,560 

900 

16,777,216 

10,0000 

60,000 

mm 

2,816 

^^00 

268,435,456 

100,0000 

70,000 

1,1170 

3,072 

(900 

3,489,767,296 

1000,0000 

80,000 

1,3880 

3,320 

800 

55,736,276,736 

1,0000,0000 

90,000 

1,5T-10 

21 


TABLE     III. 


Vulgar  Fractions^  Tonals  and  Decimals. 


Decimal. 

Tonal. 

Decimal. 

Tonal. 

1 
2" 

-  0.5 

1        0.S 

I       0.6875 

,^-0-^ 

1 

4 

-0.25 

1-0.4 

u      0.8125 

■0  -  0-^ 

1 

8 

-  0.125 

1  =  0.2 

I      0.9375 

3 

4 

-0.75 

,         0.* 

i      0.03125 

4  -  0-OS 

3 

8 

-  0.375 

^_0.b 

,^  -  0.29 16G.. 

24 

^  —  0.4999.. 

5 

8 

'-  0.625 

^  -  0.4166.. 

-  —  0.6999.. 

7 
8 

-  0.875 

^_u.^ 

^  -  0.3333.. 

0 

^  —  0.5555.. 

0 

1 
16 

-  0.0625 

i     0-1 

4  -  0.6666.. 

0 

f  —  0.9999.. 

0 

3 
16 

-  0.1875 

■^-^••^ 

\  -  0.1666.. 
0 

-  —  0.2999.. 

6 

5 
16 

=  0.3125 

^=0.5 

^-0.015625 

40  -  00* 

16 

—  0.4375 

■0 -  "•' 

^  -  0.380625 

.^-o-o* 

9 
16 

—  0.5625 

^  ".. 

:^-0.0078125 

128 

4  -  002 

22 


TABLE    IV. 

Addition  and  Sublraction.     Tonal  System. 


1 

2 

3 

4 

5 

6 

7 

8 

^ 

9 

% 

'19 

8 

S 

T 

10 

2 

4 

5 

6 

7 

8 

5 

9 

% 

19 

8 

S 

f 

10 

11 

12 

3 

5 

6 

7 

8 

6 

9 

% 

19 

8 

S 

T 

10 

11 

12 

13 

4 

6 

7 

8 

rs 

9 

^ 

19 

8 

Z 

f 

10 

11 

12 

13 

14 

5 

7 

8 

i> 

9 

^ 

19 

8 

g 

f 

10 

11 

12 

13 

14 

15 

6 

8 

n> 

9 

^. 

'19 

8 

Z> 

f 

10 

11 

12 

13 

14 

15 

16 

7 

i> 

9 

1^- 

19 

8 

S 

T 

10 

11 

12 

13 

14 

15 

16 

17 

8 

9 

^ 

19 

8 

Z 

T 

10 

11 

12 

13 

14 

15 

16 

17 

18 

n> 

^^ 

to 

8 

g 

f 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Id 

9 

19 

8 

g 

f 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Id 

19 

% 

g 

g 

T 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Id 

19 

1^ 

19 

^ 

T 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Id 

19 

1^ 

119 

8 

T 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Id 

19 

1^ 

It 

18 

g 

10 

11 

12 

13 

14 

15 

16 

17 

18 

Id 

19 

1^ 

1(9 

18 

IS 

f 

11 

12 

13 

14 

15 

16 

17 

18 

16   19 

1^ 

119 

18 

12  IT 

10 

'12 

13 

!l4 

15 

16 

17 

18 

Id 

19  1^^ 

Vt 

18 

1% 

IT  20 

TABLE    V. 

Midtiplication  and  Division.     Tonal  Si/stem. 


1 

2 

3 

4 

5 

6 

7 

8 

d 

9 

% 

10 

8 

2 

T 

10 

2 

4 

6 

8 

9 

19 

S 

10 

12 

14 

16 

18 

19 

It^ 

12 

20 

3 

6 

d 

'19 

T 

12 

15 

18 

1^- 

12 

21 

24 

27 

29 

28 

30 

4 

8 

'(9 

10 

14 

18 

110 

20 

24 

28 

2i^ 

30 

34 

38 

319 

40 

5 

9 

T 

14 

Id 

IS 

23 

28 

28 

32 

37 

310 

41 

46 

A^ 

50 

6 

10 

12 

18 

U 

24 

29 

30 

36 

31^ 

42 

48 

42 

54 

59 

60 

7 

Zo 

15 

I'LO 

23 

29 

31 

38 

3T 

46 

48 

54 

5^. 

62 

6d 

70 

8 

10 

18 

20 

28 

30 

38 

40 

48 

50 

58 

60 

68 

70 

78 

80 

d 

12 

1^ 

24 

28 

36 

3T 

48 

51 

59 

63 

m 

75 

72 

87 

do 

9 

14 

IZ 

28 

32 

319 

46 

50 

59 

64 

62 

78 

82 

8'(9 

d6 

90 

% 

16 

21 

210 

37 

42 

48 

58 

63 

62 

7d 

48 

8T 

d9 

95 

^0 

t 

18 

24 

30 

319 

48 

54 

60 

610 

78 

84 

do 

319 

98 

^4 

190 

8 

19 

27 

34 

41 

U 

5^ 

68 

75 

82 

8f 

dio 

99 

%Q 

1^3 

80 

S 

11^ 

29 

38 

46 

54 

62 

70 

7S 

81^ 

d9 

98 

^6 

m 

82 

20 

T 

IZc 

23 

319 

A% 

59 

6d 

78 

87 

d6 

95 

B4 

m 

82 

21 

TO 

10 

20 

30 

40 

50 

60 

70 

80 

do 

90 

^0 

too 

80' 20 

TO 

100 

23 


EXPLANATION  OF  TABLES. 


Table  I.  shows  the  different  notation  of  equal 
numbers  in  the  decimal  and  tonal  systems,  where  it  will 
be  seen  that  the  new  system  require  a  less  number  of 
figures  in  expressing  a  high  number;  decimal  134zi:  86 
tonal,  yet  the  real  value  is  the  same  in  both  cases. 

Table  IL  is  a  further  extension  of  Table  I.  useful 
for  transferring  numbers  from  one  system  to  the  other. 

Example  1.  Required  how  the  number  31,868  will 
be  noted  by  the  tonal  system  ?     ■ 

{  30,000  —    7550  1 
1,000  =      3S8 
800  —      320 
68  =r        44 


Decimal,   < 


>  Tonal. 


3,1868  =  7M(9 

Example  2.  The  year  1859  expressed  by  the  tonal 
system,  will  be  739,  or  it  would  apparently  carry  us 
back  over  11  centuries. 

Example  3.  A  lady  of  35  years,  required  how  old 
she  will  be  by  the  tonal  system?  The  answer  is  23 
years. 

Table  III.  is  an  excellent  illustration  of  the  utility 
of  the  tonal  system.  It  contains  the  ordinary  fractions 
used  in  the  shop  and  the  market.  It  will  be  seen  that 
the  vulgar  fractions  in  daily  use,  require  four  to  seven 
decimals,  where  the  tonal  system  require  only  one  or 
two  figures.  It  must  be  admitted  that  it  is  more 
natural  to  divide  things  into  halves,  quarters,  eighths 
or  sixteenths,  than  into  fifths  or  tenths,  and  when  the 


24 

natural  fractions  are  expressed  by  decimals,  they  be- 
come   too   complicated  for   the    ordinary   uneducated 

3 

mind,  as  —  is   equal  to   0.1875,  which   I  am  assured, 

cannot  be  conceived  by  the  very  best  arithmeticians, 
but  they  know  by  practice  in  calculation  that  it  is  so. 

3 

In  the  tonal  system  it  is  very  easy  to  conceive  that  — 
is  equal  to  0.3 

Table  IV.  is  for  addition  and  subtraction,  arranged 
in  the  ordinary  way,  that  where  the  vertical  and  hori- 
zontal columns  cross  one  another  is  the  sum  of  the 
index  numbers. 

Example  4.     3  +  5  =  8,  5  +  9  =  T,  and  t-\-Z>^l6. 

For  subtraction,  find  the  greatest  number  in  the 
column  in  which  the  smaller  number  is  the  index, 
and  the  index  of  the  cross  column  is  the  difference, 

as  17  — t^  =  ^^. 

ADDITION. 


Ex.  5.    < 


To        36^'9S 
Add     10-:)78 


I 


I^Same  47526 


2  +  8  =  16"] 
9  +  7  =  11. 
^  +  ^  =  14.  . 
6  +  0  =  6.  .  .  J^ 
3+ 1=4 


47526 


f  89T^ 
Ex.  6.    \     308 


83210 


^  +  ^  +  8  =  119  ^ 

f  +  g  +  0  =  lg     I 

9  +  5  +  3  =  12 


4  +  8  =  19 


\ 


8321'^  J 


25 


Ex.  7. 


'  3819 

T3 
0 
0 


\ 


•ST 
•01 
•34 
•03 
•49 


r  678.10 

1(1 


Ex.  8. 


3T(98^61 


945 

9f 


74012 


Ex.  1. 


SUBTRACTION 

'  From  38S9T  ^ 
Subt.     4^.13  ! 


UifF.    3431 C 


Ex.  9. 


r+  81042^ 
1  —    4210f 


Ex.  t. 


C+  89^g0-01f 
I  —    2;00f-301 


im\m  [    7(9^(90-812 

In  all    arithmetical    operations,   the  tonal  fractions 
work  precisely  the  same  as  decimal  fractions. 

Table    V   is    an   ordinary  arranged   mnltiplication 
table. 

MULTIPLICATION. 


Ex.  10. 


C     389^.6 
I  6 


154044 


80-1T9 

72 


6  X  6  =  24 

6  X  ^  =  42  . 

6X9  =  3(9.  . 

6  X  8  =  30  .  .  . 

6X3  =  12 

154044 

38^206-4f 
000684 


Ex.  8.    \      1013T4 
384586 


Ex.  Zo.    { 


3^47154 


22f81d3'(9 

1105TO3278  . 

154742589  .  . 

171^621191(9 


26 


DIVISION 


Ex.  f. 

3   189ia^^0  83-5.-U9-99 
18 

182 

Ex.  10. 

41989-1)008  2901965- 13393 
394 

09 

D 

1249  .... 
1234 

119 .  .  .. 

n.. . 

16:30  .  . 
1588  .  . 

n. . 

^80. 
95(9. 

05. 

to. 

c148 
^19 

20 

230 
182 

20 
IS 

550 

576 

20 

690 
576 

1290 
1234 

fi?on 

27 


Table 

of  Tonal  Logarithms. 

Number. 

Logarithm. 

Number. 

Logarithm. 

1 

00 

D 

'd-m 

2 

0 

4 

9 

0-84 

3 

0 

66 

% 

0-88 

4 

0 

8 

l^ 

o-2;6 

5 

0 

^5 

n 

0-^8 

6 

0 

96 

% 

0-T4 

7 

0 

14: 

T 

0-T^. 

8 

0-'l9 

10 

100 

This  table  of  ional  logarithms  is  a  good  illustration 
of  the  simplicity  of  the  system.  In  logarithms  for 
single  figures,  the  montissa  contains  only  one  or  two 
fonals,  where  the  decimal  system  has  a  tail  of  an 
endless  number  of  decimals. 


TONAL  SYSTEM  OF  WEIGHTS,  MEASURES,  AND  COINS. 

A  unit  for  the  measurement  of  length  ought  to  be 
of  a  convenient  size  for  the  artizan  in  laying  out  work. 
Units  of  about  the  length  of  the  English  foot  seems  to 
be  almost  universally  adopted,  but  it  will  be  observed 
that  the  artizan  generally  employs  two  such  units,  or  a 
two  foot  rule,  which  length  appears  to  be  the  best 
suitable  for  the  actual  operation  of  measurement.  In 
some  countries  tniits  of  about  this  length  are  employ- 
ed, as  the  Swedish  alu,  Russian  archin,  the  elle  of 
Germany,  and  others,  most  of  them  approaching  the 
length  of  about  2  feet  or  the  footstep  of  a  man.  The 
French  meter  is  the  longest  unit  employed  for  ordinary 
measurement  in  the  shop  and  the  market.  In  accord- 
ance with  my  own  observation  on  the  actual  perform- 
ance of  laying  out  work  with  the  French  meter,  and 


'2S 

also   from   comments  made  by  Frenchmen,  I  believe 
that   the  meter  is  too  lour/,  to   be   convenient  for  the 
artizan,  and  when  made  for  the  pocket,  a  great  many 
joints    are   required,  which    are    objectionable    in   its 
application.     The  meter  being  divided  into  100  parts 
makes  it  inconvenient  to  divide  the  joints,  ten  are  too 
many, — four  will  contain   the  odd   number  25  centi- 
meters or  2|  decimeter  in  each  part, — five  parts  are 
not  practical.     When  the  artizan  applies  the  meter,  he 
cannot  well  see  the  correctness  at  both  ends  without 
placing  himself  in  an  inconvenient  position,  by  which 
the  correctness  of  the  measurement  is  liable  to  error, — 
having  myself  in  Paris  been  witness  to  the  fact  alluded 
to.    The  French  have  divided  the  quadrant  of  the  earth 
into  fixed  number  10,000,000  parts  in  preference  to 
giving  the  artizan  a  convenient  unit  for  his  measure- 
ment.    The  division  of  the  quadrant  of  the  earth  is 
merely  once  a  matter  of  calculation,  and  could  easily  be 
divided   into  an   odd  number,  rather  than  to  give  the 
artizan    a    unit    which    does    not    suit   him.      If    the 
10,000,000    parts    had    some    even    relation   with    the 
general  division  of  the  earth's  great  circle,  as  to  the 
length  of  one  degree  or  minute,  it  would  have  furnished 
a  good  reason  for  the  length  of  the  meter.     The  quad- 
rant of  the  earth  divided  into   the  most  natural  or 
binary  divisions  hal/s  and   halfs,  would  lately  arrive  to 
a  length  of  about  23|  inches,  which  would  have  been 
a  much  more  suitable  unit  than  the  meter  which  is 
nearly  40  inches. 

When  a  new  unit  of  length  is  to  be  selected,  it  ought 
to  be  so  adjusted  as  to  bear  an  even  relation  to  the 
length  of  minutes  and  seconds  on  the  great  circle  of  the 
earth.     By  the  tonal  systefn  it  would  become  the  most 


'  tL-'^'^y^f.'^  '  •'^  V  \r'WVrsf^  -^         '—    ■—  ---i^     -  ,  '  ^    -..     .  .      '  -w^  ^VTX^  ^   \f^  W^ 


A-Zy-t-^ 


t^      r' //    -  '-  - 


y>v  t.^-<H.i-vy^i>  A.  /«>*"'/   A  ''-  /-M.^29 


natural  to  divide  the  circle  of  the  earth  repeatedly  by 
the  tonal  base.  The  mean  circumference  of  the  earth  is 
about  2J:851-64  miles,  or  131216659-2  feet,  which  latter 
divided  by  16  X  16  X  16  X  16  X  16  X  16  X  16  =  16" 
(1000,0000  tonal)  would  be  3,489,767,296  parts,  each 
of  a  length  of  0-48882  feet,  or  5-86584  inches.  Sup- 
pose this  to  be  adopted  as  a  unit  for  the  measurement 
of  length,  and  to  be  divided  and  multiplied  by  the 
tonal  base,  its  full  size  appearance  will  be  as  shown  by 
fig.  1.  (This  figure  is  drawn  on  the  rule  fig.  3).  Meter 
seems  to  be  a  good  and  proper  name  for  the  unit  of 
length,  and  will  therefore  retain  that  word  by  calling 
the  new  unit  the  tonal  mete?'. 

It  seems  to  me  that  the  most  correct  way  of  ascer- 
taining the  size  of  our  globe,  would  be  to  measure 
the  longest  straight  distance  on  the  land,  which  by 
examining  the  map,  we  will  find  is  in  about  31°  north 
latitude ;  starting  from  Changhae  China,  through 
Tchintou,  over  the  Himalaya  mountains,  Bassora, 
Isthmus  of  Suez,  Cairo,  Cadames,  to  Santa  Cruz,  a 
distance  of  about  7700  miles,  or  over  130  degrees  in 
longitude.  This  distance  should  be  ascertained  by 
actual  measurement,  compared  with  astronomical  ob- 
servations, and  fixed  points  located  at  every  tmton. 
The  only  obstructions,  though  not  serious,  in  this 
distance,  are,  lake  Tai-Hou  in  China,  about  40  miles, 
and  the  Himalaya  mountains. 

The  same  could  be  repeated  in  America,  from 
Washington  to  San  Francisco,  in  the  38th  parallel,  a 
distance  of  about  2240  miles,  or  about  45  degrees 
diff'erence  in  longitude.  AVhen  these  distances  are 
known  with  their  corresponding  latitude  and  longitude, 
the   great  mean  circumference  of  our  globe  is  easily 


30 

calculated  by  well  known  rules  in  mathematics.  Every 
Nation  could  by  the  same  rule  find  out  the  length  of 
the  standard  tonal  meter  in  their  own  country. 

Length. 

Tonal  System.  Old  System. 

'     m.  =1  Metermill  —  0'001432  inches  Ens. 


lOUO 


-1-  m.  =  1  Metersan  =  0-022913 

luu 

1   m.  =  1  Meterton  =  0-366615 

10 

One  Meter  =:  5-86584  in.=:0-48882  ft. 

10  meter    =z  1  Tonmeter  z=  7-82112  feet. 

100  meter    =  1  Sanmeter  =125-135  " 

1000  meter    —  1  Millmeter  —  2002-207 

It  will  be  perceived  that  when  the  word  meter  is 
placed  before  the  expression  of  value,  it  impresses  on 

the  mind   a  fraction,  as  meter-mill  zn  ^^-  or  r-rr  of  a 

mill  lUUO 

meter;  and  when  the  expression  of  value  is  placed 
before  the  unit,  it  denotes  a  multiplication  of  the  same, 
as  Tonmeter  rz  10  meter. 

The  minute  measurements,  as  wire  and  needle  gauges, 
to  be  tonally  numbered  and  divided.  In  the  present 
Birmina'ham  wire  o-auo;e  the  hio^hest  number  denotes 
the  smallest  dimension,  which  ought  to  be  the  reverse. 
Suppose  the  meterton  to  be  divided  into  100  tonal  parts 
(256  decimal)  or  metermills,  each  would  be  about  i  of 
No.  36  B.  W.  gauge,  or  0-001432  of  an  inch;  this  part 
to  be  noted  No.  1  and  the  meterton  would  be  No.  100 
which  is  about  f  of  an  inch.  By  such  arrangement, 
the  very  expression  of  the  number  impresses  the  mind 
of  the  real  size  of  the  minute  measure,  derived  from 
the   main    standard,  the  circumference  of  the   earth 


31 

Such  a  gauge  would  most  likely  be  generally  adopted 
for  minute  measurements  in  shops  where  the  present 
B.  W.  gauge  was  never  known. 

The  tonal  meter  to  be  employed  in  manufactories, 
for  measuring  machinery,  &c.,  corresponding  to  the 
English  foot.  The  artizan  generally  carries  a  two  foot 
rule,  folded  into  two  or  four  parts,  the  tonal  measure 
would  be  of  precisely  the  same  shape,  but  with  four 
units  instead  of  two. 

On  the  accompanying  plate  are  full  size  drawings  of 
the  tonal  measure,  of  which  fig.  2  is  a  four-folded  rule 
of  one  meter  in  each  part,  in  appearance  very  much  like 
ordinary  four-folded  two  foot  rule.  The  side  A  A  con- 
tains the  meter  tonally  divided  and  numbered.  The 
other  side  B  B  of  the  same  rule,  contains  divisions  for 
circumference  and  areas  of  circles,  arranged  so  that 
opposite  the  diameter  on  A  is  the  circumference  on  B 
and  area  on  C.  Suppose  the  diameter  to  be  I'tQ  meters 
on  A,  it  corresponds  with  5-48  meters  the  circumfer- 
ence on  B,  and  2-5  square  meters  the  area  on  C.  The 
small  divisions  between  B  and  C  are  each  four  meter- 
sans  drawn  from  A  to  assist  the  transference  and  read- 
ing on  B  and  C. 

Fig.  3  represents  a  two-folded  tonal  measure, 
similar  to  the  English  two-folded  two  foot  sliding  rule, 
numbered  and  divided  same  as  fig.  2.  The  part  E  on 
which  fig.  1  is  drawn,  to  receive  numbers  of  specific 
gravity  of  substances,  and  other  co-efiicients  of  general 
nse  in  practice.  The  scales  F,  G,  and  H,  are  the  ordi- 
nary sliding  rule,  divided  into  the  tonal  system,  which 
in  this  case  stands  in  such  relation  to  the  divisions  on 
the  side  D  D,  that  any  number  on  H,  corresponds  with 
its  logarithm  on  D.     The  operation  on  the  tonal  slide 


32 

rule  will  be  the  same  as   that  on  the  ordinary  decimal 


one.* 


The  clear  and  simple  relation  between  numbers  aiid 
logarithm  in  the  tonal  system  has  led  me  to  some  valua- 
ble conclusions  in  reference  to  calculating  machines, 
and  mathematical  instruments,  which  I  believe  would 
be  of  the  greatest  service  to  the  world. 

The  Tonmeier,  (7-82112  feet)  to  correspond  with  the 
Fathom,  to  be  used  for  measuring  ropes,  cables,  depths 
of  water,  &c.,  &c.  The  Smimeter  (125-135  feet)  to  be 
the  length  of  the  surveying  chain,  to  consist  of  100 
(256  decimal)  links  of  one  7netcr  each. 

The  Millmeter  (2002-207  feet)  for  road  measure  and 
distances  at  sea,  to  correspond  with  miles.  One  mill- 
meter  is  equal  to  one  timmiU.,  see  division  of  time. 
Longer  distances  on  the  earth's  surface  would  be 
expressed  in  Tims. 

Astronomical  distances  would  be  best  to  express  in 
great  circles  of  the  globe,  by  which  the  mean  distance 
to  the  sun  would  be  ^fl  circles.  Great  distances, 
such  as  to  fixed  stars,  could  be  easier  conceived  by  this 
measure. 

Time  and  the  Circle. 

Tonal  System.  Old  System. 

One  circle  =10  Tims  —  24  h'rs  or  360  degrees. 

1  Tim         -  10  timtons      =    IJ  "  22°30' 

Itimton     zulOtimsans      -     5"  37i'-        l°24'22i" 
1  timsan     =  10  timmills     =  21-1=-  5'    9" 

1  timmill   -    1  Millmeter  -  1-31836^'^'=°'^'^^-         19-77" 

The  length  of  a  pendulum  vibrating  timmills  will  be 
%'666  meters  =  67-975  inches. 


*  A  few  tonal  measures  are  now  being  made  in  Philadelpliia. 


33 

The  time,  circle,  and  compass  would  thus  be  equally 
divided,  and  greatly  simplify  all  astronomical  and  nau- 
tical tables  and  calculations. 

In  expressing  time,  angle  of  a  circle,  or  points  on 
the  compass,  the  unit  tim  should  be  noted  as  integer, 
and  parts  thereof  as  ional  fractions^  as  5*86  Urns  is 
live  times  and  mcionhy.  The  unit  Urn  to  be  pro- 
nounced as  in  the  English  word  t'miber. 

The  accompanying  figures  are  drawings  of  a  clock 
or  watch  dial,  and  a  compass  on  the  tonal  sysiem. 

Fig.  4  shows  the  appearance  of  a  ional  clock  dial, 
the  time  indicated  is  9-3?.,  which  expressed  by  words 
should  be  Kotim  and  titonhu.  The  tim  hand  goes 
round  only  once  in  a  night  and  day,  being  on  0  at 
midnight,  and  on  S  at  noon.  If  a  third  index  hand 
is  added  on  the  same  centre,  to  represent  the  second 
hand  on  our  ordinary  watch,  it  should  make  one  turn 
on  the  dial  for  each  timsan,  when  the  small  division 
on  the  circle  would  indicate  ilmlongs  or  y  o|^-o  ^^  part  of 
the  thn^  which  is  y-||o  parts  of  our  present  second. 
Such  delicate  measures  of  time  are  often  required  in 
Physical  Science,  as  in  Astronomical  observations, 
velocity  of  light  and  electricity,  gunnery,  &c.  A 
further  extension  of  delicate  measures  of  time  will  be 
conceived  in  musical  vibration,  which  I  shall  arrange 
into  immediate  connection  with  the  tonal  watch,  that 
the  base  note  for  the  natural  key,  shall  make  10 
(16  dec.)  vibrations  per  timmill.  Turn  yourself  towards 
the  south  with  the  tonal  watch  in  vour  hand,  and  it 
will  be  found  that  the  timhand  follows  the  sun  nearly; 
or  lay  the  Watch  horizontally,  so  that  the  timhand 
points  tow^ards  the  sun,  and  the  figures  on  the  dial 
will  give  0  north,  8  south,  4  east  and  F  west,  nearly. 


34 


1  io"  4 


This  is  easily  comprehended  by  the  public,  as  the 
tonal  compass,  fig.  5,  is  divided  the  same  way.  A 
course  noted  from  the  tonal  compass  is  clear  and 
simple. 

One  miUmeter  in  length  on  the  equator  corresponds 
with  one  timmill  in  time.  By  this  division  of  time,  it 
is  always  clear  whether  it  is  in  the  morning  or  even- 
ing, without  any  special  notation.  Our  present  system 
often  leads  to  error  or  confusion,  whether  a  noted 
time  is  meant  in  the  morning  or  evening. 


35 

Fig.  5. 


Division  of  the  Eartli's  great  Circle. 

The  latitude  or  meridians  should  be  divided  from 
north  to  south  into  8  tms^  with  0  at  the  north  pole,  4 
tims  at  the  equator,  and  8  at  the  south  pole.  The 
equator  to  be  divided  same  as  the  clock  or  compass. 

Nations  ought  to  agree,  to  count  the  longitude  from 
one  meridian  drawn  through  a  fixed  point  on  the 
globe.  The  different  notation  of  longitude  on  maps  is 
a  great  inconvenience  and  sometimes  causes  confusion. 
In  my  present   traveling  I  have  maps  on  which   the 


36 

longitude  is  noted,  some  from  Greenwich,  some  from 
Paris,  Pulkova,  Washington,  Ferro,  and  on  some  maps 
it  is  not  stated  from  where  the  longitude  is  counted. 
Independently  of  the  different  points  from  where  the 
meridians  are  noted  on  maps,  the  present  divisions  of 
the  circle  make  it  very  complicated  to  calculate  the 
difference  of  time  between  places,  and  very  few  will 
understand  how, — in  fact  the  complication  is  such  as 
to  discourage  many  persons  from  the  attempt ;  while, 
if  the  circle  and  time  were  divided  as  herein  proposed, 
the  very  figures  denoting  the  meridians  would  give 
the  difference  of  time  by  simple  subtraction. 

In  the  Canary  Islands  appears  to  be  a  proper  point 
to  place  the  zero-meridian,  as  the  ancient  geographers 
who  liave  taken  their  first  meridian  from  the  west  side 
of  the  Island  of  Ferro  17°  52'  west  from  Greenwich. 

Maps  constructed  on  such  principle,  would  to  our 
descendants  forever  indicate,  not  only  the  true  posi- 
tion of  the  place  on  our  globe,  but  the  scale  of 
latitude  would  give  all  distances  on  the  maps  in  miles, 
(timmills)  feet  (meters)  and  inches  (metertons)  also 
the  area  in  acres  ;  and  a  difference  of  latitude  placed 
along  a  parallel,  would  give  the  correct  distance 
corresponding  with  time  in  longitude.  Those  plain 
matters  are  by  our  present  system,  not  only  compli- 
cated in  calculation,  but  are  seldom  thought  of,  for 
the  complication  screens  away  the  simple  knowledge. 

Measure  of  Surface. 

Tonal  System.  Old  System. 

One  square  meter  rr  0"239       square  feet. 

1  Square  tonmeter  rz  61'15  " 

1  Square  sanmeter  r=  15658'768        " 

1000  Square  meters  =  0-36  Akres. 


37 


The    square  sanmeter  to  be  the  measure   of  hind, 
corresponding  to  the  acre. 


Measure  of  Capacity. 

Toual  Systrra. 

1  Gallsan  =  10  Gallmills  = 


1  Gallton  =  10  Gallsans 
1  Gall      =    1  Cub.  Meter 


1  Tongall  =  10  Galls 


on  System. 

0*79  cub.  in.  about  a 
table  spoon. 

12  62  cub.  in.  about 
a  tumbler. 

201-78  cub.  in.  about 
a  gallon. 

11  Bushel. 

about  30  cub.  feet. 

478-2  cubic  feet. 

17-75  cubic  yards. 


1  Sangall  =10  Tongalls  — 

1  MilLall=:  10  Sanjralls  = 

1  Millgall  ■=     1  cub.tonmeter  = 

The  Gall  or  cuhic  meter  to  be  the  unit  for  measures 
of  capacity,  in  ordinary  market  practice.  The  Scmgatl 
to  be  the  measure  of  excavation  and  embankments, 
also  for  grain,  &c.  The  Millgall  to  be  the  measure  of 
firewood,  being  one  c}it)ic  tonmcter. 

Measure  of  Weight. 

One  cuhic  meter  of  distilled  water  will  weigh  7'3017'4: 
pounds  avoirdupois,  to  be  the  tonal  unit  for  weights, 
and  to  be  called  a  Pon. 


Toniil  Sj'stem. 


1  Ponmill 
1  Ponsan 
1  Ponton 
1  Pon 
1  Tonpon 
1  Sanpon 
1  Mill  pon 


10  Ponmills 
10  Ponsans 
10  Pontons 
10  Pons 
10  Tonpons 
10  Sanpons 


Old  System. 

=  0*0284:8  drams  avoi. 

r=  0--45568 

=  0  45568  pounds    " 

=  7-3017 

=  116  8 

-=  1868-8  lbs.  =  0-838  tons. 

=  13-34  tons. 


38 

The  pressure  of  the  atmosphere  will  be  about  46 
jwns  per  square  metef\  and  the  height  of  a  column  of 
mercury  balancing  the  atmosphere,  about  5  meters. 

The  force  of  gravity  will  cause  a  body  to  fall  3-:5-2?'7 
meters  in  the  first  timmill  in  a  vacuum,  and  the  end 
velocity  will  be  72  562  meters  per  timmill. 

The  Ponsan  to  be  the  unit  for  apothecary  and 
minute  weights.  Pon  for  the  ordinary  market  prac- 
tice. Sanpon  as  shipping  unit  and  heavy  weights, 
corresponding  with  the  ton. 

Measure  of  Power. 

One  pon  lifted  one  meter  in  one  timmill,,  to  be  called 
one  effect.  By  the  present  system,  one  pound  lifted  one 
foot  in  one  second  is  called  one  effect,  of  which  there  are 
550  effects  on  one  horse-power  or  55  effects  on  one 
man's-power. 

Tonal  System.  Old  System. 

One  effect  z=  2-704  effects. 

i  man's-power  zz.  10  effects  =  43'268  eff,=i  0-86  man. 
1  horse-power  =:  10  men     zz  692*3    eff.i^  T25  horses. 

The  mai'Cs'poiver  to  be  the  unit  for  manual  labour, 
and  horse-power  for  machinery  and  hea^'y  work. 

Money. 

The  American  dollar  is  nearly  the  mean  difference 
of  all  the  monetary  units  of  the  world,  and  curious 
enough,  compared  with  the  largest  the  English  pound 
sterling  £,  and  the  smallest,  the  French  Franc  F,  the 
Dollar  D,  will  be  the  mean  proportion  of  the  two. 


39 

L  :  D  =  1)  :  F.  or  D  :=  YTV. 

If  the  world  could  agree  to  adopt  one  unit  for 
money,  it  seems  that  the  dollars  has  a  claim  to  be 
chosen  as  a  standard. 

Tonal  System.  Old  System. 

One  dollar  =10  shillings  =  One  dollar  =100  cent. 

1  shilling    =  10  cents        =  6:5  cents. 

1  cent  =  0-39  cent    =  2  centims. 

The  inconvenience  of  the  monetary  decimal  system 
is  daily  felt  in  the  actual  market  practic,  for  although 
the  dollar  is  divided  into  100  parts,  for  which  suitable 
coins  (most  of  odd  numbers  of  dollars  and  cents)  are  in 
circulation,  the  retail  prices  of  most  articles  are  fixed 
to  suit  the  dollar  divided  into  16  parts.  A  drink  of 
almost  any  description  costs  6  or  6J  cents,  which  is  Jg 
part  of  a  dollar.  A  ride  in  an  omnibus  costs  generally 
6  cents.  Suppose  an  article  bought  for  6  cents,  and 
paid  with  a  quarter  of  a  dollar,  there  will  be  19  cents 
change,  summed  up  by  the  following  coins  lOct.  +  5ct. 
+  3ct.  +  let.  r=  19  cents.  This  can  reasonably  be 
called  an  odd  system  of  calculation,  because  there  is 
nothing  but  oddity  about  it.  By  the  to7ial  money, 
the  same  article  paid  by  a  quarter,  which  would  be  4 
shillings,  there  would  be  3  shillings  change,  in  which 
transaction  the  mind  was  carried  only  to  4,  while  the 
decimal  money  w^as  fumbling  about  among  the  odd 
numbers  up  to  25. 


40 


Tonal  Clin?. 


United  States  Coin 


f      1  cent. 
Copper,   {      2  cents. 
I      4  cents. 


Silver, 


Gold, 


8  cents. 

1  shilling. 

2  shilling. 
4  shilling. 
8  shilling. 
1  dollar. 


1  dollar. 

2  dollars. 
4  dollars. 
8  dollars. 

10  dollars. 
20  dollars. 


Copper,  ^      1  cent 


Silver, 


Gold, 


3  cents. 

5  cents. 

10  cents. 

15  cents. 

20  cents. 

25  cents. 

50  cents. 

100  cents. 

1     dollar. 

3    dollars. 

2i  dollars. 

5    dollars 

10    dollars 

20    dollars 

The  tonal  coins  are  all  of  even  and  of  the  easiest 
countable  numbers,  such  as  are  required  in  the  market, 
while  the  decimal  coins  are  most  of  odd  numbers,  and 
of  a  complicated  composition  for  calculation,  even  the 
half  dollar  or  50  cent  has  a  prime  number  to  its  index. 
The  ional  coins  give  a  nicety  of  a^-Jg th  part  of  a  dollar, 
while  the  decimal  coins  give  it  only  to  j^-^  part. 

The  legal  interest  on  money,  in  most  countries  is 
about  6  per  cent,  which  by  the  tonal  system  would 
be  nearly  10  per  sant ;  consequently,  calculating  that 
interest  on  money,  would  be  only  to  point  off  two 
figures  on  the  capital. 

If  the  tonal  interest  is  1  more  or  less  than  10  per 
sant,  it  is  calculated  by  simple  addition  or  subtraction. 


41 


Interest  on  3^5c^4  dollars  at  10  per  sant.  =  3^5-':54 


(C 


(( 


(( 


(C 


a 


11 


ii 


=   3T0-^g4 


which  is  3T0  dollars,  &  shillings,  and  8-  cents. 

This  makes  a  simple  interest  calculation  in  the 
neighbourhood  where  it  is  most  wanted.  The  differ- 
ence of  1  "pev  cent  interest  in  the  neighborhood  of  6, 
is  a  rather  large  margin,  for  which  we  often  find  it 
accompanied  with  a  fraction,  in  practice.  One  per 
sant  tonal  z=z  O'o91  per  cent  decimal.  One  decimal 
per  cent,  is  2*56  per  sant  tonal.  Fractions  would  be 
rarely  required  to  the  percentage  in  the  tonal  system. 

The  most  common  retail  prices  of  articles  in  Amer- 
ica are  as  follow : 


Market  Prices  ct. 

Tonal  Shillings  or  16ths  of  a 
Dollar. 

Nearest  Decimal  Cents. 

6J 

1 

6 

12i 

2 

12  or  13 

m 

3 

19 

25 

4 

25 

31J 

5 

31 

371 

6 

37  or  38 

43f 

7 

44 

50 

8 

50             ' 

56J 

6 

56 

62i 

9 

62  or  63 

m 

« 

69 

75 

19 

15 

81i 

8 

81 

87J 

S 

87  or  88 

93| 

f 

94 

100 

10 

100 

42 

It  may  be  remarked  that  those  prices  are  retained 
from  the  circulation  of  Spanish  Coins  in  the  United 
States,  to  which  I  beg  to  reply  that  if  such  prices  and 
coins  were  not  the  most  natural  to  the  mind,  and  the 
most  suitable  for  the  market  they  would  not  be 
retained. 

Postage  Stamps. 

The  following  are   the  Post   stamps  of  the  United 

States. 

let.,  Set.,  lOct.,  12ct.,  24ct.,  30ct.,  90ct. 

The  very  first  glance  at  this  series  shows  plainly 
that  there  is  some  confusion  about  it.  The  stamps  of 
even  post  prices  are  not  even  in  a  dollar  (except  let.) 
and  four  of  them  are  not  even  in  any  coin,  there  is  a  10 
cent  post  stamp  and  no  10  cent  postage.  The  simple 
and  even  numbers,  most  valuable  in  calculation,  as  2, 
4,  8  and  16,  are  of  necessity  omitted,  because  the 
decimal  system  does  not  admit  the  natural  numbers. 
Let  us  now  turn  our  attention  to 

Tonal  Post  Stamps. 


Tonal  Stamps.                                       Ameru 

an  Cents. 

4  cents        for  city  post 

—    ly\  cents. 

8  cents         "    single  letters 

-    31       " 

1  shilling     "   double  letters 

-    6i       " 

2  shillings    "   quadruple  letters 

-  12i      " 

4  shillings    "      8 

—  25 

8  shillings    '^10 

—  50 

1  dollar        "    20 

—    1  dollar. 

Here  it  will  be  found  that  the  tonal  pod  stamp  series 


43 

contains  the  even  number  most  simple  for  calculation, 
and  they  are  even  both  in  post  prices  and  in  the  tonal 
coins  or  dollar. 


Division  of  the  Year. 

The  new  year  ought  to  commence  at  Christmas,  and 
the  year  divided  into  10  (16  deci.)  months,  which 
would  make  about  17  (23  deci.)  days  per  month. 


S 

E 


a 


o-a 

3 


1 

2 
3 
4 
5 
6 
7 
8 
;S 
9 
% 
10 
8 

10 


'"  a 


17 

16* 

17 

16 

17 

17 

17 

17 

17 

17 

17 

17 

16 

17 

17 

17 


Names  of  the  new 
months. 


Anuary. 

Debrian. 

Timander. 

Gostus. 

Suvenary. 

Bylian. 

Ratamber. 

Mesudius. 

Nictoary. 

Kolumbian. 

Husamber. 

Vyctorious. 

Lamboarv. 

Polian. 

Fylander. 

Tonborius. 


The  first  day  of  the 

new  month 

to  commeocu  on. 


21  December. 
13  January. 

4  February. 

27  February. 
21  March. 

13  April. 
6  May. 

29  May. 
21  June. 

14  July. 

6  August. 
29  August. 
21  September. 
13  October. 

5  November. 

28  November. 


New   year   and 
Christinas. 


\  Night   and   day   of 
^     equal   length. 


I  Midsummer  day,  or 
St.  John. 


\  Night    and  day    of 
^     equal   length. 


*  17  Days  in  Leap  Year. 


There  will  be  168  tonal  days  in  a  year,  and   16S  in 
leap  years. 


44 

The  names  of  the  tonal  months  are  given  so,  that  the 
first  syllable  expresses  the  number  of  the  month  in 
the  year,  and  every  four  months  have  a  similarity  in 
sound,  impressing  the  quarters  of  the  year.  The  new 
year  and  Christmas  should  be  on  the  same  day. 
There  is  no  occasion  for  altering  the  days  in  the  week, 
but  when  days  are  to  be  expressed  by  tonal  fractions 
of  the  months  or  year,  the  number  of  days  are  nearly 
80  per  sant  more  than  the  fraction,  for  instance  6  days 
=r  0-4  months  or  0-04  years,  6  days  =  0*6  months, 
19  days  r=  0-8  months,  T  days  =:  0-09  years,  12  days  = 
O-OIO  years,  &c.,  &c.  The  different  Kalenders  used  in 
different  Countries,  would  by  the  tonal  system  at  once 
fall  into  one.  The  old  or  Julian  style  is  yet  used  in 
Russia  and  other  Countries,  it  is  12  days  behind  our 
new  or  Gregorian  style.  The  Evangelistic  year  com- 
mences December  27,  on  the  day  of  St.  John ;  this 
style  is  also  adopted  in  Freemasonry,  where  it  is  known 
as  the  Masonic  year.  The  tonal  sfyle  would  become 
seven  days  ahead  of  the  Gregorian. 

Measure  of  Heat. 

The  three  different  thermometrical  scales  causes  a 
great  deal  of  inconvenience  in  science  and  art.  Al- 
though Fahrenheit's  scale  is  generally  employed  in 
the  United  States,  yet  we  have  American  Scientific 
books  in  which  Celcius'  scale  is  used  exclusively. 
Celcius'  decimal  scale  is  the  most  convenient  for 
calculation,  but  I  believe  that  those  degrees  are  too 
large  for  scientific  purposes,  that  we  want  the  scale  to 
be  divided  into  more  parts  between  the  freezing  and 
boiling  points. 


45 

By  the  tonal  system  it  would  become  tlie  most 
natural  to  divide  the  thermometer  scale  into  100  (256 
decimal)  parts  between  the  freezing  and  boiling  points 
of  fresh  water. 

Tonal  System.  Old  System. 

Zero  or  0  =  +  82        Fah.  or  0    Celcius. 

1  Temp  =r  10  tempton  =z       11 J      Fah.  or  6 J 

1  Tempton  =  0-7    Fah.  or    0-4    " 

Tenq)  for  rough  measurement  of  the  temperature  of 
the  weather,  and  tempton  for  scientific  purposes. 

3Iu8ic. 

The  many  different  clefs  used  in  music,  seems  to  be 
a  complication  without  remuneration.  Music  Corps 
often  use  four  different  clefs,  namely,  Bass^  Tenoi\ 
Treble  and  Alto^  all  of  which  could  be  of  one  single 
denomination.  The  Tenor  and  Alto  clefs  are  gradually 
withdrawn,  but  there  is  yet  no  indication  of  dispensing 
with  the  Bass  or  F.  clef  In  Piano  music  particularly, 
it  is  an  unnecessary  complication,  and  burdens  the 
student,  to  have  a  different  denomination  on  each 
stave.  I  shall  here  arrange  it  so  that  all  the  different 
clefs  will  be  represented  in  one  denomination. 

The  standard  pitch  of  tone,  I  will  assume  to  100 
(256  Arabic)  vibrations  per  timmill,  for  the  base  note 
in  the  natural  key.  As  A  is  the  first  letter  in  the 
alphabet,  it  appears  natural  that  it  should  be  the  first 
or  base  note  in  the  natural  key,  and  that  such  an 
octave  from  A  to  A,  should  be  located  within  the 
musical  stave. 


46 

I  would  propose  to   denote   five  different  clefs,  as 
follows : 


CANTO 
CLEF. 


Soprano. — This  clef  to  run  two  octaves 
above  the  treble  pitch. 


ALTO  CELF. 


Contralto. — One  octave  above  the  treble, 
for  high  female  voice. 


TREBLE 
CLEF. 


XI  Descant. — Natural  position,  or  ordinary 
Y-      female  voice. 


TENOR 
CLEF. 


For  the  common  voice  of  man,  one  oc- 
tave below  the  treble. 


BASS  CLEF.     - 


The  ordinary  bass,  two  octaves  below 
the  treble  pitch. 


In  running  music,  where  the  notes  extend  too  high 
above,  or  too  deep  below,  the  stave,  the  octave  can  be 
altered,  by  placing  the  suitable  clef  on  the  bar  where 
the  change  is  required,  similar  to  that  now  employed 
as  8!i. 

The  following  natural  scales,  in  the  different  clefs, 
show  the  number  of  standard  vibrations  per  timmill, 
of  each  note,  divided  according  to  the  geometrical 
progression  or  tempered  scale,  as  employed  in  practical 
music : 


47 


800 

8<f8    914-4  99{^-y  gfyS  g74-4  T19-4  1000 

- 

-  .    _    ::  _   .  ___^^     -^               ^      a      r  T 

CANTO  CLEF. 

i^— 

-^-     -9    --| —     r-     -f—     ' 

i 

J      • 

»                      1        ,           ■    ■  ■ 

A 

40C 

B          C         D         E          P          G         A 
)     478-8  509-2  556-2  5Ti;-4  6«9-2  78e-2    800 

-^ 

- 

1             1                        a          9         P 

A  LTO  CLEF 

-( 

\ 

1       1      ^      #      r            1 

-A 

jr 

# 

V                                          1               , 

B 


D         E  F  G 


TREBLE 
CLEF. 


i; 


200     23S-y  285-1  29S-7  2'f'f-2  25e-l  3VP.4    400 
i ^— ^- 


i: 


9 

B 


It 

C 


"I 


D 


E 


P 


G 


TENOR  CLEF. 


100     llT-5  142-4  155-y  17'f-cS  19S-8  123-4    200 


'-^-^_ 


If: 

B  C 


9- 

D 


"I         r" 
'I 

E  P 


G 


BASS  CLEF. 


80 

8'f-9    91-4     99-S    %'i-V    £7-4    Tl-9    100  "^ 

— \ 

\- 

II                    m        <*       ^ 

\       '\      ^      9      r             1 

m 

*      f      *      1        '.               ' 

-^       9          "                                    1        ,    J.      .    ...                  \ 

A  BCDEFGA 


The  natural  key-note  A,  vibrating  100  (256  Arabic) 
per  timmill,  will  be  194  per  second,  which  corresponds 
nearly  with  Y^  in  the  high  pitch  now  used.  Com- 
plaints have  often  been  made,  that  our  present  pitch 
of  music  is  too  high,  and  it  has  been  proposed  to  lower 
it  down  to  256  vibrations  per  second  of  C,  as  accepted 
in  acoustics,  when  the  tonal  pitch  of  A  would  corres- 
pond nearly  with  the  present  G,  by  which  the  com- 
parative cromatic  scales  would  be  as  follows: 


48 


TONAL 
TRIPLE. 


:i 


B 


1 — r — \: 


C     D 


I      I 


E 


-#•- 


i=*-s*-F^^-F-^f 


G    A 


ORDINARY 

TRIPLE. 


-t-l'- 


1       ^' 


G 


:^-#fzpip=Er=*?EE 


B    C 


E 


F     G 


By  this  arrangement,  the  present  key  of  G  should 
be  the  natural  tonal  key.  The  notes  on  the  musical 
stave  would,  in  the  tonal  system,  bear  the  same  names 
as  in  the  old  Bass  clef;  but  the  distance  between  C 
and  D  would  be  only  half  a  note,  also  that  between  G 
and  A,  as  shown  on  the  tonal  scale. 

The  advantage  of  this  arrangement  would  be  to 
attain  a  universal  standard  pitch,  and  to  have  only  one 
simple  denomination  in  all  music. 

Examining  the  nature  of  fingering  on  most  musical 
instruments,  we  will  find  that  the  key  of  G  is  most 
naturally  located,  as  in  the  Flute,  Clarionet,  Violin, 
Guitar,  and  many  others. 

Brass  instruments  ought  to  be  arranged  so  as  to  give 
the  natural  chord,  without  using  the  keys  or  pistons. 


1 

f 

- 

— ^ 

^-T-^-f-^- 

- 

..       #              1 

The  key-board  on  the  Piano  should  be  moved  down 
five  half  notes,  for  the  same  strings  to  suit  the  tonal 
music. 

The  different  keys  in  sharps  and  flats  would  appear 
as  follows : 


49 


E    MAJOR, 


^;jt- 


"]' 


B    MAJOR 


■I 


"1' 


'~r 


9 


F    MAJOR. 


-#- 


^ 


C    MAJOR. 


D   MAJOR. 


C^^lfc 


G   FLAT  MAJOR. 


-b— i- 


C    FLAT  MAJOR. 


fcg: 


±: 


:# — r-zjzi 


-  &c.,&c. 


Of  all  the  different  sciences,  that  of  music  seems  to 
be  the  most  neglected;  and,  shame  to  say,  it  is  yet 
under  consideration,  if  music  shall  be  admitted  as  a 
science.  It  would  be  out  of  place  to  treat  on  the 
science  of  music  in  this  book ;  but  as  I  have  already 
touched  the  subject,  I  will  finish  by  making  a  few 
remarks. 


50 

The  best  performers,  who  have  natural  talent  for 
music,  does  not  require  the  science  of  the  same ;  but 
for  composers,  it  is  of  great  importance,  and  for 
musical  instrument  makers,  it  is  indispensable.  One 
reason  why  musical  taste  is  so  little  cultivated,  is  for 
the  want  of  good  instruments,  and  the  whole  world  is 
filled  up  with  musical  instruments  of  inferior  quality, 
for  the  want  of  science  in  their  manufacture.  Many 
of  them  make  perfect  cat  music,  wlien  it  is  unreason- 
able to  expect  of  the  performer  to  cultivate  musical 
taste.  I  have  often  wished  to  put  down  such  musicians 
to  the  grade  of  organ-blowers,  but  reflecting  on  the  sub- 
ject, I  find  myself  greatly  in  error — it  is  the  instrument 
maker  who  is  to  be  blamed,  and  not  the  performer. 

The  only  wind  instrument  constructed  on  purely 
scientific  principles,  and  which  has  attained  perfection, 
is  the  Boehm  Flute,  which,  I  believe,  is  the  best 
example  to  show  what  science  can  do  in  music.  Mr. 
Boehm  himself  is  an  excellent  performer  on  the  Flute, 
he  is  a  good  composer,  a  skillful  mechanic,  Avitli  inven- 
tive ingenuity,  and  he  is  a  scientific  man.  Every  one 
of  these  faculties  are  brought  to  bear  on  the  construc- 
tion of  his  Flute,  and  it  could  not  have  been  perfected 
without  one  of  them. 

In  the  year  1858, 1  had  the  pleasure  of  Mr.  Boehm's 
acquaintance,  in  Munich,  when  he  was  constructing  a 
G  Flute,  which  is  2|  notes  below  our  ordinary  C  Flute. 
Mr.  Boehm  remarked,  that  the  finest  musical  ear 
cannot  detect  a  difference  in  a  tone  so  delicate  as  he 
can  establish  it  by  calculation  and  measurement.  I 
examined  very  closely  the  details  of  the  holes  and  keys 
on  the  G  Flute,  which  was  made  of  silver,  and  found 
that  no  allowance  was  made  for  retoning  it  by  trial 


51 

and    error,    but   it   was    made    right   from    the   first 
design. 

The  finest  musical  wind-instrument  in  Existence,  I 
believe,  is  the  Bassoon,  (Fagott) ;  but  for  the  want  of 
science  in  constructing  it,  it  is  now  very  much  declin- 
ing. If  a  Bassoon  was  constructed  under  similar 
treatment  as  that  of  Boehm's  Flute,  I  believe  it  would 
produce  a  sound  approaching  the  finest  human  voice. 
The  tones  on  the  Bassoon  sound  very  much  like  one 
singing  in  the  nose,  which  is  a  clear  indication  that 
the  musical  vibration  is  jammed  up  at  the  point  where 
it  is  broken  ofi". 


Abreviation  of  the  Tonal  Units. 

M  =  Metei\  unit  for  length. 

G  _  Gall,  unit  for  capacity. 

T  -~  Tim,  unit  for  time  and  the  circle. 

P  =  Pon,  unit  for  weight. 

H  —  Horse-poiver,  unit  for  work. 

D        Dollar,  unit  for  money. 

Tp  ^  Temp,  unit  for  temperature. 

The  abridgment  of  the  units  to  be  noted  by  capital 
letters,  and  the  multiplication  and  division  of  the  same 
as  an  exponent  by  a  small  letter  placed  before  or 
after  the  unit,  thus,  M'  z=.  Meterton,  'M  rz  Tonmeter, 
G'  =r  Gallsan,  P  Timsan,  P'"  =:  Ponmills,  &c.,  &c. 

The  sound  of  the  new  names  will  of  course  be 
strange  to  the  ear,  before  its  corresponding  character 
and  value  is  known  by  heart,  but  as  before  stated,  the 
object  aimed  at  has  been  to  select  such  sounds  as 
would  become  best  suited  for  all  languages,  and  at  the 
same  time  be  simple  and  expressive. 


52 


Example  11.     When  3-109P  of  butter  cost  4-38D, 
how  much  will  1^-22P  of  the  same  cosf? 
•         3-1^9  :  1^-22  =  4-38  :  X 
X  =  ^^■^<:_\^-^^  -  12-36  dollars. 


3- 1-9 


1^^-22 
4-38 

8^70 
5189 

619^8 

72-9910 


3-^9  I  72-9910 
31^9.  .. 


12-36 


3609  . 
35019 . 


852. 

1830 

16^19 

« 

174 

The  answer  is  12  dollars,  3  shillings  and  6  cents. 

Example  12.  What  will  be  the  interest  on 
32(98 -65D  in  4  years  and  5  months,  at  9  ])ev  sant 
per  annum  ]     (9  per  sant  is  about  6  per  cent,  decimal.) 

Interest  r=  321^8-65  X  009  X  4.-5  =  953-82  dollars. 

Example  13.  A  yearly  payment  or  annuity  of 
328-65D  is  standing  for  15  years  and  S  months, — what 
will  it  amount  to  in  that  time  at  ^-  per  sant  interest  1 

*  .  o-^o  ^r    V  X    -i  -i   Q)     r  1       I    0-OB    /i-i.aj      ,      INT    


884T-82  dollars. 

328-65 

15-^ 

xju  y\   jLtj  iLi    L  i  -p 

119-^ 
0-0058 

5473-127 
1-58198 

244357 
135888 
32865 

258 
8T7 

29358538 
3T564284  . 
445g72T^ 

0-58K8 

2T50^95T 

5473-127 

5473127 

884^-8296178 

53 

Example  14.  AVhen  one  gall,  of  fresh  water  weighs 
one  pon,  how  much  will  5-8S  cubic  meters  of  cast-iron 
weigh,  when  the  specific  gravity  of  the  iron  is  7'212'? 

Weight  7-212  X  5-8S  =  2^-846319  Pons,  the  answer. 

A  similar  example  by  the  old  system  will  be  very 
complicated.  Even  in  the  French  metrical  system 
there  is  more  confusion  in  pointing  off  the  decimals. 

Example  15.  A  locomotive  running  89 •^'"M  per 
Tim,  leaves  London  at  5-g4T ;  another  locomotive  on 
the  same  track  makes  6S"K™M  per  Tim,  leaves  London 
at  6*2feT.  At  how  many  Millmeters  from  London, 
and  at  what  time  will  the  faster  locomotive  reach  the 
other  1 

The  time  of  the  fast  locomotive  will  be, 

i  zr  -—z — :r '  z=  0-7  tims,  the  answer, 

6  2?;— 39  s  '  ' 

and  6-2^^  +  0*7  =:  G'^'B  tims,  the  time  when  the  fast 
locomotive  reaches  the  other.  Distance  from  London 
will  be  6'^-\9  X  0-7  =  30-74  Millmeters. 

Examples  in  Navigation,  Comjmring  the  Old  and  Tonal 

Si/stems. 

Example  16.  In  the  year  1861  the  sun's  declina- 
tion at  Greenwich  mean  noon  is : 

Old  System.  Tonal  System. 

March  13,     2°  48'  36-9"     —     0-lTf9  tims. 
March  14,     2°  27'  57-3"     =     0-1^78      " 


Difference,         23'  39-6"     =:     0-0478      " 


Required  the  true  declination  at  mean  noon,  on  the 
13th  of  March,  1861,  in  longitude  west  from  Green- 
wich, 156°  40'  23"  =  6-T70d  tims  =  0-6T70^  days. 


54 


0-4353 

23  Mult. 

Old  System. 

156° 

60  Mult. 

360 
60  Mult 

13059 

8706 

9360                     21600  Mill. 
42  Add.                  60  Mult 

100119  Mill. 
0-0119 

^      9402  Min.        1296 
60  Mult. 

000  Sec. 

60  Mult. 

564120 

23  Add. 

0-7140 

Add  39-6  Seconds. 

564143  Seconds. 

40-3140 

40  314 
0-4353 

5641430000 
5184000  .  .  . 

129600C 

0-4353 

120942 
201570. 
1 20942 

4574300  .  . 

38880U0  .  . 

161256... 

6863000  . 

17-5486842  Seconds. 

3830000 
3888000 

from  -which  the  correction  will  be  10'  17-548". 

Tonal  System. 

DifF.  longitude  0-6T70d  days. 

DifF.  declination  0-0478  tims. 


5^.0^75 
361013T 
1^8(924 


Correction       0-01f934365  tims. 


55 

The  required  declination  will  be, 

2°  48'  36-9"  =  OlTf  9  tims. 
Correction,  add     10'  17-5"  =  001T9     " 

True  decli.         2°  5S'  54-4"  =  0-21f4  tims,  the  answer. 

The  old  system  required  seven  multiplications,  one 
division,  four  additions,  and  employed  in  the  calcula- 
tion about  215  figures,  while  the  tonal  system  required 
only  one  multiplication,  and  employed  only  39  figures 
for  the  same  object,  namely,  to  find  the  correction. 

The  old  system  required  a  great  deal  more  know- 
ledge of  how  to  manage  the  many  different  operations 
and  figures,  and  in  consequence  subject  to  more  errors, 
while  the  tonal  system  is  simple,  clear,  and  natural  to 
the  mind.  In  workins:  the  time  and  lunar  observations, 
the  difference  will  be  still  more,  on  account  of  angles 
being  expressed  both  in  time  and  degrees. 

Although  our  decimal  arithmetic  is  based  on  10,  the 
ordinary  mind  has  found  it  more  simple  to  divide  their 
units  into  8,  12,  16,  20,  &c.,  &c.,  parts,  but  as  the 
science  in  arithmetic  advanced,  the  educated  mind 
found  that  the  units  could  also  be  divided  into  the 
unsuitable  number  10  parts,  and  so  the  complication 
was  carried  into  weight  and  measure. 

Had  the  tonal  system  been  adopted  instead  of  the 
decimal  arithmetic,  it  would  surely  be  considered 
ridiculous  to  divide  units  into  10  parts. 

In  the  measurement  of  machinery  there  are  very 
few  dimensions  that  come  up  to  the  length  of  the 
French  metei\  then  most  of  the  measurement  must  be 
expressed  in  decimal  fractions,  often  with  several 
cyphers  before  the  figures  as  is  readily  seen  on  French 
drawings.     A  French  millimeter  O-OOl'"-  zi:  0-039  inches. 


56 

A  tonal  santimeter  O'Ol™-  z=  0*0229  inches.  Here  the 
French  system  require  one  cypher  more  in  expressing 
a  quantity  58  per  cent,  greater  than  the  tonal  ex- 
pression, and  the  difference  will  be  much  more  in 
squares  and  cubes. 

In  calculating  the  weight  and  cubic  contents  of 
machinery  we  have  often  very  small  dimensions  to 
deal  with,  then  a  cubic  centimeter  will  be  0-00000 1*"-  ™- 
which  by  the  tonal  system  will  be  only  O-OOl"-  ""■  or  three 
cyphers  less  for  the  same  dimension.  Decimal  frac- 
tions of  this  kind  make  more  or  less  confusion 
by  the  many  cyphers,  in  squaring  or  cubing  numbers, 
and  more  so  in  extracting  roots. 

The  French  metrical  system  is,  however,  not 
uniform  throughout,  by  which  it  is  self-evident 
that  the  meter  is  too  large.  The  imit  for  ca- 
pacity one  litre  =r  O'OOl  cubic  meter,  and  the  unit 
gramme  =  0  000001  cubic  meter  of  distilled  water 
in  Aveight. 

In  the  tonal  system  there  are  no  such  irregularities. 
One  cubic  meter  is  the  unit  for  capacity,  and  the 
weight  of  one  cubic  meter  of  distilled  water  is  the 
unit  for  weight. 

It  is  self-evident  that  the  French  metrical  system  is 
not  well  suited  in  practice,  for  although  it  has  been 
enforced  by  law  in  France  for  over  twenty-three  years, 
yet,  expressions  of  the  old  system  are  frequently  used 
in  the  shop  and  the  market ;  and  oftentimes  particu- 
larly in  the  interior,  bargains  are  made  in  the  old 
system  and  the  bills  made  out  in  the  new  system.  In 
the  Paris  market,  it  is  most  invariably  heard  that  an 
article  cost  so  many  sous  per  lime^  and  it  is  very  natural 
that  it  remains  so,  because  the  decimal  system  is  too 


57 

troublesome.  It  is,  of  course,  easier  to  count  twenty 
sous  on  the  franc  instead  of  one  hundred  centimes. 

In  dividing  things,  it  often  happens  that  10  parts 
will  be  too  fine  divisions,  or  the  delicacy  of  the  work 
may  not  require  all  the  10  parts  ;  it  is  then  suggested 
to  take  only  every  other  part,  but  in  so  doing  it  does 
not  fall  in  with  half  the  base  5,  for  which  it  may  be 
necessary  to  reject  either  five  or  none  of  the  divisions, 
which  is  a  great  inconvenience  in  practice.  On  ther- 
mometer scales,  we  often  find  every  other  degree 
marked,  which  does  not  come  in  with  the  fives. 

In  the  tonal  system,  every  other  or  every  four  parts 
will  fall  in  with  the  halfs  and  quarters,  as  w^ell  as  with 
the  base. 

THE   COUKTING   MACHIKE. 

The  counting  machine,  fig.  6,  consists  of  a  square 
frame,  in  which  are  inserted  ten  brass  wires  or  lines  a 
he  —  h,  eight  of  which  have  each  10  tonal  balls,  move- 
able from  one  side  to  the  other.  The  void  line  c,  is  to 
denote  tonal  fractions^  that  is,  the  balls  on  the  line  «, 
to  denote  cents,  metersans,  timsans,  &c.,  &c.,  and  the 
balls  h  for  shillings,  metertons,  galltons,  &c.,  &c.,  d  the 
unit,  e  10,  /  100,  g  1000,  i  10000  and  k  100000.  The 
line  /z,  separates  the  mill  and  hong,  to  make  the  read 
ing  more  clear. 

Before  the  operation  of  counting  is  commenced,  all 
the  balls  are  to  be  on  the  left  side,  and  to  be  moved 
towards  the  right  as  the  counting  requires.  The  in- 
strument is  to  be  used  principally  for  addition  and 
subtraction,  but  multiplication  and  division  can  also  be 
performed  on  it,  with  some  assistance  of  mental  calcu- 
lation. 


58 


X  1^. 


6. 


00 


MttOOOOl^ 


illDIIMOOCO 

QQDQc 


jpfinn/ 


lOOOOMHOOOfr 


The  operation  on  the  counter  is  similar  to  that  on 
the  ancient  abacus. 

The  number  noted  on  the  counter  is  74^-26,  which 
according  to  whatever  unit  it  m.eans,  may  be  74^-  dol- 
lars, 2  shillings  and  6  cents,  of  which  700  is  on  /,  40 
on  e,  and  ^'  on  d.  Suppose  the  sum  of  1345-42  dollars 
is  to  be  added  to  74^-2.JD.  Move  one  ball  to  ^,  which 
denotes  1000,  3  balls  towards/,  4  towards  e,  5  towards 
J,  4  towards  h,  and  2  towards  a  ;  and  the  sum  will  be 
found  on  the  left  side  l96QQt  dollars.  The  operation 
is  generally  commenced  on  the  top  line  «,  and  when 
there  are  not  balls  enough  on  the  line,  subtract  the 
complement  of  10  and  add  one  on  the  next  line  below. 
This  will  be  understood  by  the  following  example : 

Add  39  cents  to  the  2'S  in  the  noted  sum.  Say  9 
from  10  is  6,  move  6  from  a,  and  there  will  be  3  left, 
move  one  towards  h^  move  further  3  balls  towards  I, 
and  the  result  will  be  63  cents,  or  6  shillings  and  3 


59 

cents,  the  sum  of  2-:)  4-39.  Add  further  1^8  dollars  to 
74^.  Say  8  from  10  is  8,  move  8  balls  from  d  and  1 
to  e ;  say  10  from  10  is  4,  move  4  balls  from  ^,  and  1  to 
/,  and  the  sum  will  be  74^.  +  m  —  813  dollars. 

In  this  manner,  addition  can  be  continued  on  the 
counter  up  to  1,0000,0000  tonal,  or  53,736,276,736 
decimal. 

By  the  aid  of  this  instrument,  the  tonal  system 
would  be  easily  acquired,  because  it  turns  the  mind 
from  the  old  base,  which  is  of  the  greatest  importance ; 
besides,  the  counter  would  be  a  most  valuable  instru- 
ment for  adding  columns  of  figures. 

THE  RUSSIAN   STCHOTY,  {Fig.  7.) 

This  instrument  is  in  common  use  in  Russia.  Every 
counting-house,  office,  store,  or  shop,  of  whatever 
description,  and  every  family  has  a  stchoty ;  in  fact,  it 
is  as  common  in  Russia  as  a  spoon  or  knife.  In  the 
steps  of  southern  Russia,  where  a  house  is  rarely  to 
be  found,  and  where  it  is  difficult  to  find  anything  to 
eat  or  drink,  a  stclioty  is  always  to  be  found,  even 
among  the  Kalmuks ;  and  it  is  surprising  to  see  with 
what  readiness  and  correctness  the  Russians  use  this 
instrument  for  their  calculations.  They  can  multiply 
and  divide  with  great  facility  on  it. 

The  operation  on  the  stchoti/  is  the  same  as  that 
described  for  the  counter.  The  line  a  denotes  quarters 
of  copeks,  h  single  copeks,  c  10  copeks,  d  25  copeks 
or  J  of  a  ruble  for  each  ball,  e  rubles,  /  1 0  rubles, 
g  100  rubles,  h  1,000  rubles,  and  i  10,000  rubles.  The 
sum  noted  on  the  stclioty  is  743-45  rubles  or  743  rubles 
and  45  copeks. 

The  line  d  is  not  absolutely  necessary,  as  the  copeks 


62 


Letter  from  the   International  Association. 

10  Farrar's  Building,  Temple,      > 
London,  31st  October,  1859.   \ 
Sir: — 

Your  esteemed  letter  of  the  1st  of  June,  has  been 
received,  together  with  a  copy  of  your  description  of 
the  Calculator,  which  is  evidently  a  most  ingenious 
and  useful  instrument, 'and  the  manuscript  account  of 
your  new  system  of  arithmetic  and  measures,  weights 
and  coins. 

We  think  ourselves  much  honored  by  the  confidence 
which  you  have  manifested  towards  us,  and  are  of 
opinion  that  we  shall  best  testify  our  high  sense  of 
your  ability  and  intelligence,  and  your  zeal  for  the 
improvement  of  mankind  by  the  most  free  and  sincere 
expressions  of  our  sentiments  upon  your  project. 

The  project  has  evidently  the  great  merit,  which,  as 
far  as  we  know,  belongs  to  no  other  hitherto  invented, 
except  the  metrical  system,  that  is  a  uniform  system 
founded  upon  one  simple  principle,  which  is  con- 
sistently applied  throughout  to  the  attainment  of  its 
professed  design.  We  are,  nevertheless,  sorry  that  we 
cannot  give  it  our  support,  having  by  the  very  consti- 
tution of  our  society,  and  from  its  first  foundation, 
adopted  the  number  10  as  the  basis  for  such  a  system. 
On  reviewing  the  grounds  of  our  original  determination 
we  see  no  reason  to  depart  from  it. 

When  the  metrical  system  was  invented  by  the 
careful  deliberation  of  the  first  mathematicians  of  the 
age,  they  studied  the  question  by  the  arithmetical  scale, 
and  especially  to  take  12  as  a  basis,  because  that  num- 
ber seemed  to  have  in  some  respects  a  claim   to  be 


63 

taken  in  preference  to  10.  After  a  full  examination  of 
the  question  they  decided  that  it  was  necessary  to 
retain  10  as  the  basis  in  arithmetic,  and  to  adopt  it 
universally  for  measures,  weights  and  coins. 

Your  system  would  be  far  more  difficult  to  learn 
than  the  other.  When  learnt,  it  would  require  a 
smaller  number  of  figures  in  each  operation,  and  might 
therefore  present  some  facilities  for  making  calcula- 
tions in  writing,  but  it  would  be  very  burdensome  to 
the  memory  so  as  to  be  unsuitable  for  mental  arith- 
metic, and  consequently  for  all  the  smaller  dealings  of 
the  shop  and  the  market,  and  for  those  minute  calcu- 
lations which  in  all  arts,  trades  and  manufactures  often 
requires  to  be  performed  with  the  greatest  possible 
rapidity.  There  is  a  limit  to  the  powers  of  the  human 
mind,  and  it  appears  probable  that,  except  in  extraor- 
dinary cases  a  system  founded  on  16  as  a  basis,  would 
be  found  to  exceed  the  natural  capacity  of  man  for  the 
use  of  numbers. 

You  object  to  the  meter,  as  "  much  too  long  to  be 
convenient  to  the  artizan,  and  you  therefore  choose  for 
your  unit  a  length  which  is  about  the  seventh  part  of 
a  meter.  The  proper  aim  in  determining  upon  a  unit 
of  length  is  to  find  one  adopted,  as  far  as  possible,  to  all 
uses  without  exceptions,  and  the  general  consent  of 
mankind  seems  to  point  to  the  conclusion  that  a  length 
approaching  to  the  meter  best  corresponds  with  this 
intention.  For  specific  purposes  the  meter  is  divided 
or  multiplied  either  by  2  or  by  5,  and  thus  you  may 
obtain  any  measure  you  please  including  your  own 
unit  which  is  very  nearly  equal  to  15  centimeters. 

We  could  show  you,  if  we  had  the  pleasure  to  see 
you  here,  numerous  decimal  divisions  of  the  meter  such 


64 

as  the  measures  5,  10,  20,  25,  30,  40,  and  50  centi- 
meters. We  have  these  graduated  down  to  half  mili- 
meters,  made  of  a  great  variety  of  substances,  and  with 
considerable  difference  of  form,  either  solid  and  in  one 
piece,  or  made  to  fold  with  hinges,  or  to  be  wound  on 
roulettes  in  cases  so  as  to  be  carried  in  the  pocket  with 
the  greatest  ease  imaginable. 

Also  some  are  made  with  slides.  In  short  the  meter 
is  proved  by  experience,  an  experience  which  is  extend- 
ing every  day  over  wider  and  wider  area  of  the  earth's 
surface,  to  be  adopted  for  the  artizan  as  well  as  for 
every  other  occupation.  You  express  a  preference  for 
the  English  foot,  but  if  the  foot  has  advantages,  you 
may  take  the  foot  of  Hesse  Darmstadt  which  is  exactly 
the  fourth  part  of  the  meter.  It  is  considerably  nearer 
than  the  English  foot  to  your  own  proposed  unit,  and 
in  itself  is  unquestionably  as  convenient  for  the  artizan 
as  any  other  foot. 

In  conclusion,  we  beg  to  present  you  with  the  prin- 
cipal publications  which  have  been  issued  by  our 
branch  of  the  International  Association. 

We  would  especially  direct  your  attention  to  our 
treatise  on  the  best  unit  of  length.  In  section  VI. 
you  will  find  a  discussion  of  the  question  respecting  its 
adaptation  to  binary  division,  and  in  section  VIII.  it 
is  maintained  in  opposition  to  your  views,  that  the 
meter  may  be  employed  with  the  greatest  possible 
advantage  in  the  mechanical  arts.  You  will  allow  us 
sir,  to  indulge  the  hope  that  further  examination  of 
the  subject  may  induce  you  to  coincide  in  the  opinion 
which  we  endeavor  to  defend  and  which  is  gaining 
ground,  as  we  understand,  in  Russia,  as  well  as  in  the 
civilized  countries. 


65 

Witli  much  respect  we  subscribe  ourselves  on  behalf 
of  the  British  Branch  of  the  International  Association. 
Your  obedient  servants, 

(Signed,)     James  Yates,  F.  R.  S.,  Vice  President. 
Leone  Levi,  Resident  Secretary. 

Mr.  John  W.  Nystrom, 

Rostof  on  the  Don. 

P.  S. — Further  observations  on  the  same  topic  are 
published  in  Lord  Overstone's  questions  with  the 
answers,  London  Folio,  1857,  176,  179.  Your  letter 
and  pamphlet  were  shown  at  a  meeting  of  the  Inter- 
national Association,  at  Bradford  on  the  10th,  11th, 
and  12th  ultimo,  and  we  were  desired  to  convey  to 
you  their  thanks  for  the  valuable  suggestion  you  have 
offered. 

Leonf  Levi,  Res.  Sec. 


ludinova,  in  the  government  of 
Kaluga,  Russia,  Nov.  26,  1859. 
To   the   International  Association   for  obtaining   a 
uniform  Decimal  System  of  Measures,  Weights, 
and   Coins,  No.   10  Farrar's  Buildings,  Temple, 
London : 

Gentlemen  : 

I  have  had  the  honor  to  receive  your  favor  of  the 
31st  of  Oct'r  last,  for  which  I  beg  to  return  my  most 
sincere  thanks.  Feel  very  much  gratified  indeed  that 
your  honorable  body  considered  my  suggestions  worthy 
of  notice. 

I  hope  the  International  Association  will  bear  with 
me  for  making  some  remarks  on  your  letter,  by  which 


66 

I  have  no  other  object,  but  to  discuss  the  connection  of 
practical  and  scientific  principles,  and  sincerely  beg 
you  to  pardon  my  straightforward  expressions. 

It  seems  to  me  that  the  real  substance  of  my  project 
for  the  tonal  system  of  arithmetic,  weight,  measure, 
and  coins,  is  not  well  conceived  or  appreciated  by  the 
International  Association,  because  there  are  statements 
in  the  letters  which  are  not  exactly  in  accordance  with 
the  fact,  likely  originated  from  the  difficulty  in  con- 
ceiving a  new  arithmetical  system  with  a  new  base, 
when  the  base  10  is  impressed  on  the  mind.  You  state 
that  my  "  System  would  be  far'more  difficult  to  learn 
"  than  the  other."  Such  is  not  the  case ;  it  is  easier 
learned,  and  although  the  multiplication  table  of  the 
single  figures  is  about  two  and  a  half  the  extent  of 
that  of  the  old  system,  it  is  much  easier  acquired. 

The  only  difficulty  is,  to  turn  the  mind  from  the  old 
base  :  for  instance,  8X8  =  40,  will  appear  very  curious 
to  a  stranger,  who  is  perfectly  sure  that  8  X  ^  ==  64, 
but  when  he  knows  that  16  is  the  base  for  the  system, 
and  8  is  half  of  the  base,  he  will  easily  conceive  that 
half  of  8  is  4,  and  8  X  8  =  40. 

"  When  learned,  it  would  require  a  smaller  number 
"  of  figures  in  each  operation,  and  might  therefore  pre- 
"  sent  some  facilities  for  making  calculations  in  writing." 
This  is  of  little  or  no  importance  in  the  sense  you 
apply  it :  it  makes  very  little  diff'erence  if  one  or  two 
figures,  more  or  less,  are  written  down  on  paper,  which 
is  a  mere  mechanical  operation  compared  with  having 
the  figures  clearly  located  on  the  mind. 

Any  measure  under  8  feet  can,  by  the  tonal  system, 
be  expressed  to  a  nicety  of  a  fraction  of  a  millimetre. 


67 

or  less  than  32nds  of  an  inch,  with  only  three  figures, 
which,  in  ordinary  cases,  would  require  at  least  five 
figures  on  the  French  metre;  and  a  measure  of  up  to 
122  feet  is  expressed  to  the  same  nicety  by  only  four 
Tonal  figures.  The  utility  of  the  tonal  system  is  not 
limited  by  the  small  number  of  figures  expressing  a 
delicate  measure,  but  on  account  of  the  figures  at  the 
same  time  impressing  the  mind  of  the  natural  frac- 
tions, quarters,  eighths,  and  sixteenths,  the  principal 
utility  lays  in  the  clearness  of  the  expression.  Six- 
teenths expressed  by  decimals  will  require  four  times 
the  number  of  figures,  which  will  carry  the  mind  1000 
times  further  than  the  tonal  system.  In  astronomical 
and  nautical  calculations,  there  will  be  required  a  less 
number  of  figures,  on  account  of  dispensing  with  a 
great  number  of  tables. 

"  But  it  would  be  very  troublesome  to  the  memory, 
"  so  as  to  be  unsuitable  for  mental  arithmetic,  and 
"consequently  for  all  the  smaller  dealings  of  the  shop 
"  and  market,  and  for  those  minute  calculations  which 
"  in  all  arts,  trades,  and  manufactures  often  require  to 
"  be  performed  with  the  greatest  possible  rapidity." 

This  is  not  right,  and  it  proves  that  the  utility  of 
the  tonal  system  is  not  within  your  comprehension. 
For  mental  calculations,  the  shop  and  the  market,  it  is 
best  suited,  and  the  very  reason  why  I  have  proposed 
it.  For  mental  calculation  in  addition  and  substrac- 
tion,  the  mind  need  not  be  carried  further  than  the 
base  10,  and  in  multiplication  and  division  only  to 
100.  I  would  not  take  the  trouble  to  invent  or  pro- 
pose a  new  system  of  arithmetic  for  "  the  greatest 
"  mathematician  of  the  age,"  to  whom  it  makes  little 
or  no  difference,  if  the  base  is  a  prime  number,  for  his 


68 

ps,  qs,  and  fs  are  applicable  to  any  system  whatever ; 
neither  would  I  think  it  worth  the  while  to  alter  the 
system  of  arithmetic  for  the  small  portion  of  the  pub- 
lic who  have  to  do  with  quantities  only  by  pen  and 
ink,  for  whom  it  is  very  easy  to  find  a  suitable  system, 
and  the  decimal  system  with  10  or  100  to  the  base  will 
answer  that  purpose  fully ;  but  it  is  not  so  in  the  shop 
and  in  the  market,  where  the  natural  fractions  and 
aliquot  numbers  are  wanted,  as  well  for  mental  calcu- 
lations as  for  the  mechanical  divisions  and  proportions 
of  materials. 

"  There  is  a  limit  to  the  power  of  the  human  mind, 
"  and  it  appears  probable  that,  except  in  extraordinary 
"  cases,  a  system  founded  on  16  as  a  basis  would  be 
"  found  to  exceed  the  natural  capacity  of  men  for  the 
"  use  of  numbers."  The  decimal  system  is  the  worst 
that  ever  could  be  selected  of  the  even  numbers  in  the 
neighborhood  of  10,  eight  or  twelve  would  require  less 
capacity  of  mind.  It  is  easily  found  in  practice  that 
when  the  base  of  a  measurement  is  12  or  16,  it  is 
easier  managed  in  the  mind,  even  with  the  present 
system,  and  it  would  become  so  much  easier  if  the 
Arithmetic  had  the  same  base. 

Had  I  proposed  11,  13,  11,  15,  or  17,  to  the  base, 
the  system  would  become  more  difficult  in  a  quadruple 
or  triple  proportion  to  the  number  of  new  digits  added, 
but  16  is  quite  an  exception  to  that  supposition;  9  as 
a  base  would  be  a  great  deal  more  difficult  than  10. 

I  will  here  give  you  some  numbers  placed  in  order 
as  they  become  more  difficult  as  a  base  for  Arithmetic, 
namely,  8,  16,  12,  10,  14,  9,  11,  15,  13.  It  can  very 
readily  be  conceived  that  the  present  arithmetical  base 
is  too  small,  because,  referring  to  the  decimal  system 


69 

in  practice,  it  will  be  found  that  it  is  generally  im- 
pressed on  the  people's  mind  that  the  unit  is  divided 
into  100  parts,  by  which  quantities  are  generally  writ- 
ten and  expressed,  for,  although  it  is  at  the  same  time 
divided  into  10  parts,  it  is  seldom  used  so.  For  in- 
stance, in  France,  it  is  never  said  or  written  that  a 
measure  is  so  many  decimetres  long,  but  it  is  expressed 
in  so  many  centimetres. 

In  America,  it  is  never  said  or  written  that  an  article 
cost  so  many  dimes,  but  it  is  expressed  in  so  many 
cents.  We  have  also  in  America,  from  the  Spanish 
money,  the  dollar,  divided  into  8  and  16  parts,  which 
are  mostly  expressed  by  separate  names,  and  in  reality 
found  to  be  a  more  suitable  division  for  the  market. 
Suppose  an  article  to  cost  38  cents,  and  it  is  paid  with 
a  dollar.  Now,  the  seller  must  carry  his  mind  to  100, 
and  then  back  to  somewhere  about  70 ;  here  he  will  be 
confused  about  the  eight,  and  not  sure  if  it  will  be  cor- 
rect to  subtract  the  8  from  70 ;  but  finally  he  finds  out 
that  it  is  62  cents  to  be  returned  on  the  dollar;  the 
buyer — most  frequently  not  so  smart  in  counting  as 
the  seller — will  perhaps  say  that  there  should  be  72 
cents  change.  This  example  I  have  given  from  actual 
and  frequent  observation  in  practice.  Now,  suppose  a 
similar  example  with  English  money :  an  article  cost 
38  pence ;  it  will  be  observed  that  38  pence  is  not 
noted,  but  it  is  said  or  written  three  shillings  and  two 
pence.  Suppose  the  buyer  to  pay  for  the  article  with 
a  crown,  which  is  five  shillings.  The  seller  will  very 
likely  reply,  "  Have  you  two  pence,  and  I  will  give  you 
two  shillings'?"  or  he  may  give  the  buyer  Is.  lOd,  and 
so  the  aff"air  will  end  with  perfect  understanding ;  the 
mind  was  not  carried  above  the  base  12,  while  in  the 


70 

American  case  the  mind  was  carried  more  than  eight 
times  further,  namely,  to  100.  The  decimal  system  is 
therefore  very  troublesome  for  mental  calculation,  and 
frequently  approaches  the  "  limit  to  the  power  of  the 
human  mind,'''  which  would  be  rarely  the  case  with  the 
tonal  system.  It  will  not  be  denied  that  halfs,  quar- 
ters, eighths,  and  sixteenths  are  the  most  natural  frac- 
tions for  the  artizan,  shop,  and  market,  and  they  are 
frequently  expressed  by  decimal  fractions;  but  if  0"125 
is  shown  to  the  majority  of  the  people,  there  will  be 
comparatively  none  who  understand  the  true  meaning 
of  it;  and  if  it  is  told  to  them  that  0*125  means  J,  it 
will  be  necessary  to  explain  that  the  whole  is  divided 
into  1000  parts,  and  125  of  the  parts  is  J  of  the  whole. 
The  people  will  then  surely  reply  that  this  is  a  round- 
about way  of  doing  things,  and  that  they  are  not  willing 
to  cut  their  things  up  into  1000  parts  in  order  to  get 
it  into  eighths.  I  am  inclined  to  believe  that  among 
the  best  arithmeticians,  including  the  International 
Association,  there  are  few,  if  any,  who  clearly  compre- 
hend that  125  is  J  of  1000,  but  it  is  well  known  to  be 
so  by  practice  in  calculation.  It  is  easy  to  comprehend 
that  25  is  J  of  100,  from  which  it  can  be  conceived 
that  1  X  25  :=  12*5,  and  by  that  way  it  may  be  im- 
pressed on  the  mind  that  125  is  J  of  1000.  If  12  was 
the  arithmetical  base,  it  is  easily  conceived  that 
^  =  0'16,  but  with  the  tonal  sjstem  it  is  most  easy 
to  comprehend  that  J  =  0  2.  Therefore  the  decimal 
system  is  very  complicated  and  difficult,  as  well  for 
mental  calculation  as  for  the  artizan's  ordinary  appli- 
cation of  numbers  and  measurement.  A  high  number 
of  several  digits  must  be  managed  in  the  mind,  in 
order  to  comprehend  a  small  one  of  only  one  digit. 


71 

In  music,  the  tonal  system  is  in  full  operation ;  the 
notes  are  divided,  as  regards  time,  into  halves,  quarters, 
eighths,  &c.,  &c. 

Fig.  8. 


A  bar  of  music  is  generally  expressed  by  quarters  or 
eighths,  and  a  burden  has  generally  8  or  16  bars. 

Now,  suppose  that  a  musician  is  requested  to  divide 
his  notes,  bars  and  burdens  into  fifths  or  tenths, 
according  to  the  decimal  system, 

Fig.  6. 


thus 


r         r  p 

-^k\     /r^^    orifyouplease  rf  Jf  X 

rrrrrrrm  rrrrrrrrrr 


then  ask  the  musician  to  play  a  decimal  piece  of  music, 
and  it  will  sound  very  much  like  the  decimal  system 
introduced  in  the  shops  and  markets.  This  is  the  best 
comparison  I  can  give  between  the  tonal  and  decimal 
systems,  because,  if  the  world  was  fortunate  enough  to 
be  in  possession  of  the  tonal  system,  and  knew  nothing 
about  it,  but  requested  to  turn  the  mind  towards  the 
decimal  system,  it  would  be  much  more  awkward  to 
mankind  than  for  the  present  musician  to  do  so. 

"  You  object  to  the  metre  as  much  too  long  to  be 
convenient  to  the  artizan," — Yes ;  and  the  Interna- 
tional Association  has  given  me  a  further  proof  of  the 


72 

fact,  which  I  beg  to  explain  hereafter, — "  and  you 
"  therefore  choose  for  your  unit  a  length  which  is  about 
"  the  eighth  part  of  a  metre."  It  is  stated  in  the  manu- 
script that  the  length  of  a  step  of  a  man,  or  about  two 
feet,  appears  to  be  a  suitable  unit,  and  when  I  divided 
the  circumference  of  the  earth  with  16^  it  was  my 
greatest  wish  to  arrive  at  a  unit  of  about  20  to  28 
inches,  but  as  the  length  of  the  assumed  main  standard 
was  not  under  my  control,  I  was  obliged  to  be  con- 
tented with  the  last  quotient  5*865  inches.  It  was, 
however,  my  intention  to  propose  to  divide  the  quad- 
rant of  the  earth  with  16",  which  will  give  a  unit  of 
about  23|  inches,  but  in  order  to  follow  a  uniform  and 
unbroken  system  of  division  throughout  all  kinds  of 
measurement,  I  concluded  to  maintain  the  first  quo- 
tient of  5-865  inches  as  the  unit  for  length. 

"  The  proper  aim  in  determining  upon  a  unit  of 
"length  is  to  find  one  adopted,  as  far  as  possible,  to  all 
"uses  without  exception." 

That  is  just  the  very  object  of  my  aim,  and  it  is 
the  inconvenience  and  defalcation  of  the  decimal  and 
metrical  system,  that  has  called  on  me  to  propose 
something  better.  It  seems  to  me  that  in  making  such 
statements,  it  would  have  been  well  to  give  some  reason 
and  example  where  the  tonal  system  is  inapplicable. 

The  metrical  system  is  inapplicable  in  navigation, 
because  it  does  not  agree  with  the  degrees  and  minutes 
of  the  great  circle  of  the  earth,  which,  also  makes  some 
inconvenience  in  geographical  survey.  The  decimal 
system  cannot  well  be  adapted  for  the  division  of  the 
circle  and  the  time,  nor  can  it  be  adapted  in  music, 
which  forms  the  most  natural  conception  of  division. 
"  The  general  consent  of  mankind  seems  to  point  to 


73 

"  the  conclusion  that  the  length  approaching  to  the 
"  metre  best  corresponds  with  this  intention."  This  is 
not  correct.  If  the  table  of  foreign  measures  of  length 
is  examined,  it  will  be  found  that  the  whole  world 
points  towards  the  English  foot,  that  the  French  metre 
stands  alone  the  longest  measure,  and  that  it  is  only 
the  Persian  arshine  which  attempts  to  approach  it. 
Nations  point  towards  the  French  uniform  decimal 
system,  merely  because  it  is,  as  far  as  our  present 
arithmetical  system  permits,  in  itself  the  most  complete 
for  calculation,  but  if  the  French  had  adopted  a  shorter 
metre,  I  believe  the  system  would  have  been  picked 
up  much  sooner  by  other  nations. 

"  For  special  purposes  the  metre  is  divided  or  multi- 
"  plied  either  by  two  or  by  five,  and  thus  you  may 
"  obtain  any  measure  you  please,  including  your  own 
"  unit,  which  is  very  nearly  equal  to  15  centimetres.'' 
That  I  do  not  understand. 

"  We  could  show  you,  if  we  had  the  pleasure  to  see 
"  you  here,  numerous  decimal  divisions  of  the  metre, 
"such  as  the  measures  of  5,  10,  20,  25,  30,  40,  and  50 
"  centimetres."  This  is  a  proof  that  the  metre  is  too 
long,  and  very  likely  some  practical  mechanic  or  engi- 
neer has  made  the  same  remark,  for  which  the  metre 
is  cut  up  into  pieces,  in  order  to  show  that  it  can  be 
made  shorter.  Let  us  examine  the  pieces  one  by 
one.  A  measure  of  only  5  centimetres  is  of  little 
importance  to  the  artizan,  besides  5  is  a  prime  number, 
which  makes  the  whole  decimal  system  objectionable. 
A  length  of  10  centimetres  is  very  convenient  for 
minute  measurement,  but  too  small  for  general  use. 
Twenty  centimetres  is  a  good  measure  within  itself, 
may  be  conveniently  used  in  the  drawing-room  and  for 


74 

measures  not  exceeding  its  length,  but  for  more  than 
20  centimetres,  it  will  be  accompanied  with  an  objec- 
tionable mental  calculation.  It  is  contained  in  the 
unit  five  times,  which  is  a  perplex  number  for  the 
artizan,  because  when  the  fifth  part  is  laid  down  he 
may  be  uncertain  whether  he  has  laid  down  four  or 
five  20  centimetres,  and  when  he  looks  back  on  the 
measured  part,  he  cannot  well  conceive  the  correctness 
without  going  over  it  once  more,  Avhich  would  not  be 
the  case  if  the  measure  was  contained  four  times 
in  the  unit,  where  the  halfs  and  halves  certify  the 
correctness.  A  measure  of  25  centimetres  has  the 
only  advantage  of  being  contained  4  times  in  the 
metre,  but  within  itself  an  unsuitable  measure ;  in 
measuring  off  a  distance  between  25  and  100  centi- 
metres, it  is  accompanied  with  a  troublesome  mental 
calculation. 

Twenty-five  centimetres  are  rather  long  for  the 
pocket ;  it  must  be  folded,  but  into  how  many  parts  1 
if  folded  into  two  parts,  there  will  be  12J  in  each. 
Thirty  and  forty  centimetres  are  not  evenly  contained 
in  the  unit,  and  will,  in  practice,  be  accompanied  with 
troublesome  mental  calculation. 

Fifty  centimetres  chopped  up  into  four  or  five 
parts,  has  its  evident  disadvantages.  If  these  diff'erent 
measures  are  introduced  into  the  market,  people  will 
become  accustomed,  one  to  a  20  centimetre,  another 
to  a  25  centimetre,  and  some  select  30,  40,  or  50 
centimetres;  then  when  one  gets  hold  of  a  strange 
centimetre,  he  is  apt  to  make  a  mistake  in  his  meas- 
urement and  calculation. 

"  We  have  these  graduated  down  to  half  millimetres, 
''  made   of  a   great   variety  of  substances,  and   with 


75 

"  considerable  difference  of  form,  either  solid  and  in 
"  piece,  or  made  to  fold  with  hinges  or  to  be  wound  on 
"  rollers  in  cases,  so  as  to  be  carried  in  the  pocket  with 
"  the  greatest  possible  ease  imaginable." 

I  am  perfectly  convinced  that  the  metre  can  be 
made  convenient  for  the  pocket,  but  I  say  that  it  is 
not  convenient  for  measurement  and  mental  calcula- 
tion, and  I  am  sure  that  it  requires  a  great  many 
ingenious  contrivances  to  put  the  metre  into  a  suitable 
shape,  but  among  all  your  varieties  of  metres  and 
centimetres,  have  you  a  single  sample  which  can 
practically  be  considered  so  good  a  measure  as  the 
English  two-foot  rule  1  You  will  allow  me  to  doubt 
it.  I  have  also  in  my  possession  a  few  varieties  of  the 
metre,  but  none  which  I  consider  a  proper  measure- 
ment, and  I  have  never  seen  a  good  metre  even  in 
France,  although  I  have  made  great  efforts  to  procure 
the  best  possible.  Those  in  my  possession  are  all 
made  to  fold  into  10  parts,  made  of  ivory,  bras^■, 
fish-bone,  wood,  and  one  a  tape  to  roll  in  a  case,  but 
they  are  all  toys.  In  Marseilles  once  I  bought  a  metre 
of  the  ordinary  form,  made  of  ivory,  to  fold  into  10 
parts,  went  home  to  my  hotel  and  tested  the  metre  on 
my  standard  rule,  and  found  it  to  be  1|  millimetres  too 
short.  I  returned  immediatelv  to  the  instrument 
maker,  Mr.  Santi,  No.  6  Ferreol  Street,  stated  the  fact, 
which  was  soon  testified  on  a  standard  metre,  and  I 
was  offered  to  select  a  correct  one,  which  made  me  try 
a  great  many  metres  one  by  one,  and  did  not  find  two 
of  the  same  length.  I  then  suspected  the  great  many 
joints,  tried  several  by  pushing  and  drawing,  when  I 
found  a  little  motion  in  some  of  them,  tried  again  two 
metres,  the  shorter  one  I  stretched  a  little,  when  it 


76 

became  the  same  length  as  the  other.  I  selected  one 
metre  by  the  standard  in  the  store,  which  I  have  now 
on  my  table ;  it  has  grown  two  millimetres  longer  when 
I  stretch  it  out,  and  when  I  push  all  the  joints  in  an 
opposite  direction  the  metre  will  be  J  millimetre  too 
short, 

I  do  not  blame  the  workmanship  of  the  metre  in 
question,  because  it  is  made  as  good  as  it  can  be,  and 
it  is  equally  good  as  those  I  selected  in  Paris,  where  I 
found  similar  metres  to  those  in  Marseilles;  but  I 
object  to  the  principle  of  the  instrument,  because  it  is 
in  every  shape  inconvenient  in  practice.  It  is  very 
inconvenient  to  lay  out  work  by  the  ordinary  pocket 
metre ;  for  instance,  the  metre  must  be  kept  and 
adjusted  by  the  left  hand  at  a,  and  stretched  by  the 

Fig.  9. 
a 


2r 


right  hand  at  ^,  it  is  then  required  a  third  hand  to 
straighten  the  decimetres  between  a  and  5,  because  the 
work  is  oftentimes  such  that  if  the  hand  is  taken  from 
J,  that  end  of  the  meter  will  fall  down,  and  disturb  the 
adjustment  at  a. 

In  practice  it  often  happens  that  it  is  inconvenient 
to  get  at  one  of  the  points  between  which  a  measure 
is  wanted,  a  two  foot  rule  is  then  stuck  over  to  the 
furthest  or  otherwise  inaccessible  point ;  and  the  meas- 
ure read  at  the  nearest  point ;  in  a  great  many  such 
cases  of  daily  occurrence,  it  would  be  impossible  to 
employ  direct  a  ten-folded  French  metre,  for  which  the 
two  hands  are  required,  one  at  each  point.  Suppose 
the  outside  diameter  of  a  cylinder  is  to  be  measured, 
it  is  generally  taken  in  a  pair  of  callipers,  then  by  the 


77 

English  mode  the  callipers  are  kept  in  the  left,  and 
the  rule  in  the  right  hand,  while  the  diameter  is  read ; 
now  by  the  French  measurement  two  hands  must  be 
employed  to  keep  the  metre  while  a  second  person 
must  be  employed  to  keep  the  callipers.  You  will 
now  surely  remark  that  "  the  metre  can  be  made  to 
"  fold  with  hinges  into  four  parts — similar  to  the 
"  English  rule,  and  used  with  the  same  advantage" — to 
which  I  beg  to  reply  that  the  metre  in  such  a  form 
will  be  rather  clumsy  for  the  pocket,  and  for  the 
artizan,  and  on  account  of  its  great  length  it  will  not 
have  the  firmness  of  an  English  rule.  Two  and  a  half 
decimetre  or  the  odd  number  of  25  centimetre  in 
each  part  is  an  indication  that  there  is  something 
wrong  about  it.  Half  a  metre  folded  into  two  or  four 
parts  is  a  broken  up  half  thing — I  say  broken  up, 
because  two  parts  will  contain  each  an  odd  number 
of  divisions  25,  and  four  parts,  will  contain  each  12 J 
centimetres. 

Another  measure  generally  employed  as  a  standard 
by  architects,  city  surveyors,  in  machine  shops,  &c., 
&c.,  about  8  to  12  feet  long,  and  very  likely  in  the 
office  of  the  London  City  Survey  will  be  found  stand- 
ards of  10  feet,  which  is  a  very  convenient  measure 
for  a  great  many  out-door  works ;  a  measure  of  that 
kind  will  be  about  three  to  four  metres,  which  are 
very  inconvenient  numbers — accompanied  with  extra 
calculations  in  laying  out  a  long  measure  for  which  a 
tape  or  a  chain  is  not  correct  enough.  If  such  a 
measure  is  made  five  metres,  it  will  be  rather  long 
and  inconvenient,  and  accompanied  with  a  mental 
calculation  which  by  the  prime  number  5  gives  an 
odd  number  at  every  other  operation. 


78 

A  measure  of  10  metres  cannot  well  be  employed 
in  the  streets,  except  in  the  form  of  a  tape  line  or  a 
chain,  but  for  such  form  10  metres  is  too  short.  A 
tape  line  or  a  chain  ought  to  be  about  50  to  100  feet 
long.  Further  you  state  that,  "  in  short,  the  metre  is 
"  proved  by  experience,  an  experience  which  is  extend- 
"  ing  every  day  over  wider  and  wider  area  of  the 
"  earth's  surface,  to  be  adopted  for  the  artizan  as  well 
"  as  for  every  other  occupation."    This  is  saying  much. 

Has  experience  ever  had  anything  to  do  with  the 
length  of  the  metre  from  its  very  first  origin  1  It  was 
according  to  your  statement  '■'invented  hj  ilie  first 
'•'mathematicians  of  the  age^'  after  which  it  was  in- 
truded on  the  French  artizan  by  law,  from  which 
experience  in  using  it.  was  necessarily  attained.  The 
mathematician  had  the  measure  of  a  quadrant  given 
to  him  in  figures,  which  he  found  was  easiest  to  divide 
by  10s  in  order  to  arrive  at  a  small  number,  but  had 
the  mathematician  been  set  into  practice  to  divide  a 
quadrant  or  a  straight  line  by  a  pair  of  compasses,  he 
might  have  discovered  that  the  most  easy,  and  the 
most  correct  divisions  are  attained  by  dividing  it  into 
half  and  halves,  which  would  have  given  a  quotient 
of  about  23  J  inches  as  a  metre.  The  length  of  the 
metre  has  nothing  whatever  to  do  with  the  utility  of 
the  French  uniform  decimal  system  of  weight,  measure 
and  coin.  Had  a  shorter  metre  been  adopted,  and  such 
a  cubic  metre  of  distilled  water  called  a  killograra,  the 
same  advantage  would  have  been  attained.  The  prin- 
cipal difficulty  in  introducing  the  decimal  system,  and 
the  general  discord  of  weight  and  measure  throughout 
the  world,  is  caused  by  the  unsuitable  base  in  the  arith- 
metical system. 


79 

It  would  indeed  be  a  great  service  to  mankind,  if 
some  leaders  of  the  scientific  world  would  deviate  a 
little  from  their  determination  to  maintain  and  pro- 
pagate a  system  so  ill-suited  for  the  purpose  for  which 
it  has  so  long  toiled. 

"  You  express  a  preference  for  the  English  foot, 
"  but  if  the  foot  has  advantages,  you  may  take  the  foot 
"  of  Hesse  Darmstadt  which  is  exactly  the  fourth  part 
"  of  a  metre."  The  Darmstadt  foot  would  do  me  pre- 
cisely the  same  service  as  any  other  of  the  nearly  35 
different  foots  employed  in  different  parts  of  the  world, 
I  do  not  care  how  many  times  it  is  contained  in  a 
metre,  because  that  has  nothing  to  do  with  the  subject 
in  question.  The  object  aimed  at  is  to  invent,  propose 
and  introduce  to  the  world  a  system  of  calculation, 
weight,  measure  and  coins,  which  would  without  excep- 
tion fulfil  all  requirements  of  mankind,  and  when  such 
is  attained,  it  makqs  no  difference  how  many  times  the 
Hesse  Darmstadt  foot  goes  in  a  metre. 

I  thank  you  most  sincerely  for  your  kindness  in  offer- 
ing to  me  the  principal  publications  issued  by  your 
branch  of  the  International  Association  and  hope  soon 
to  receive  them,  and  I  shall  read  them  with  the  greatest 
interest.  I  am  well  convinced  that  you  have  furnished 
plenty  of  good  materials  in  favor  of  the  decimal  and 
metrical  system,  which  are  current  among  a  great  many 
of  your  readers.  You  can  give  many  instances  where 
the  metre  can  be  conveniently  employed ;  you  can  give 
examples  that  when  so  many  tons,  cwts.  and  pounds  is 
multiplied  by  so  many  £  sterling  shillings  and  pence, 
and  divided  by  so  many  fathoms,  feet  and  inches,  will 
be  a  long  and  complicated  calculation,  compared  with 
the  measures  at  once  expressed  by  decimals ;  besides 


80 

the  metrical  and  decimal  system  being  adopted  and  in 
successful  operation,  by  one  of  the  first  empires  in  the 
world,  is  indeed  a  great  temptation. 

According  to  your  statement,  I  expect  to  find  in 
"  Section  VIII.,  it  is  maintained  in  opposition  to  my 
"views  that  the  metre  may  be  employed  with  the 
"  greatest  possible  advantage  in  the  mechanical  arts." 
Such  is  easily  maintained  in  writing,  but  go  to  prac- 
tice, and  give  an  English  mechanic  a  French  metre 
of  the  ordinary  ten-folded  form,  and  ask  him  to  measure 
a  distance  of  about  20  inches ;  the  mechanic  will  then 
fumble  about  in  straightening  the  decimetres,  and  if 
there  is  no  support  between  the  two  points,  he  will 
hang  the  metre  in  a  catenary  form,  as  he  is  not  accus- 
tomed to  employ  two  hands  for  such  a  small  measure, 
— he  will  then  very  likely  tell  you  that  the  measure 
between  the  points  is  50  and  some  small  marks  which 
he  cannot  read.  Now  give  an  English  2  foot  rule  to 
a  French  mechanic,  to  measure  a  similar  distance,  and 
he  will  tell  you  immediately  without  hesitation  that 
it  is  20  and  f . 

The  inches  being  divided  into  halves,  quarters  and 
eighths,  makes  the  reading  so  clear,  that  the  very  first 
glance  impresses  the  mind  of  the  correct  measure. 

A  captain  sailing  along  the  sea  cost  in  a  dark  night, 
requires  to  be  if  possible  always  in  sight  of  a  light, 
in  order  to  be  sure  of  his  position,  and  safety  of  his 
ship ;  such  is  the  case  with  the  mind  sailing  along  a 
graduated  measurement.  On  the  English  rule,  fig.  ^', 
there  are  big  lights  at  short  intervals,  and  beacons  and 
buoys  between  them,  while  in  the  wilderness  on  the 
French  metre,  fig.  1^,  you  encounter  sometimes  a  little 
light  high  up  in  the  arithmetic  atmosphere,  looking  very 


81 


Fig.  U. 


7 


III' 


8 


I  I  I 


±j_^ 


9 


much  like  the  old  street  oil  lamps  before  lighting  gas 
was  invented,  and  between  them  you  encounter  a  num- 
ber of  things  one  like  the  other,  by  which  you  are  not 


Fig.  V. 


m 


a  5 


sure  whether  you  are  here  or  there.  The  ordinary 
English  rule  such  as  made  by  Mr.  Elliott,  London,  or 
Field  and  Son,  Birmingham,  will  stand  and  measure  as 
long  as  twenty  French  pocket  metres,  and  it  will 
measure  the  last  piece  as  correct  as  the  first  one,  which 
is  not  the  case  with  a  ten-folded  metre  stretched  a  few 
times. 

On  mathematical  instruments  in  general,  the  deci- 
mal division  is  very  troublesome,  compared  with  the 
natural  divisions,  for  instance,  in  verniers,  fig.  8,  this 
makes  a  clear  reading,  and  divides  the  inch  into  256 
parts,  while  the  decimal  system,  fig.  ^,  is  more  difficult 
and  divides  the  inch  into  only  100  parts. 

Fig.  8.  Fig.  S. 


1 1 1 1 1 1 1 


1 1 


j_ 


TT 


7 


\  I 


/ 


1 1 1 1 1 1 1 1 1  III  1 1 1 


b: 


I  I  1  I 


.1  I  I  I 


I  I  I  I 


i~rr 


1 


j\ 


I  regret  very  much  to  say  that  the  closer  I  examine 
the  subject,  the  more  I  am  inclined  to  oppose  the 


82 

French  metre,  as  well  as  the  decimal  system,  which 
is  in  reality  the  most  unnatural  system  of  division, 
which  could  reasonably  be  selected. 

I  am  sure  that  in  a  thorough  practical  examination 
the  metre  will  stand  a  poor  chance,  and  I  shall  be 
much  mortified  if  the  law  intrudes  upon  me  such  in- 
convenient measurement  for  my  mechanical  works. 

Weight. 

The  decimal  system  is  equally  inconvenient  for 
weight  as  for  all  other  measurements,  the  unit  being 
divided  into  10  parts,  for  which  are  required  -five 
different  weights  in  weighing  all  the  ordinal  parts 
namely  1,  2,  3,  5,  and  10,  or  a  weight  of  4  may  be 
substituted  for  the  3,  but  it  is  at  any  rate  an  odd  and 
dreary  composition  of  weights. 

1  =  1  weight. 

2  =  2  weights. 

3  =  3       " 
3+1=4       " 

5  =  5  " 

5+1=6  " 

5+2=7  " 

5+3=8  « 

and     5  +  3  +  1=9  " 

thus  all  the  ordinal  parts  of  10  can  be  weighed.  Now 
suppose  a  similar  example  with  the  tonal  system, 
which  will  also  require  five  weights,  namely,  1,  2,  4,  8, 
and  10,  this  is  the  most  natural  composition  of  weights, 
they  are  convenient  in  the  operation  of  weighing  and 
easy  for  mental  calculation. 


83 

1  1=  1  -weight. 

2  n:  2  weights. 

2+1=3 

4  —  4       « 

4  +  1  =:  5 
4  +  2  =  6 
4+2+1=7 
8  =  8 
8  +  1  =  .1 
8+2=9       " 
8  +  2  +  1  =  ^       « 
1  +  4  =  '(9       " 
8  +  4+1  =  8'    " 
8+4+2=2 
and       8  +  4  +  2  +  1  =  T       " 
thus  all  the  ordinal  parts  of  10  (16)  can  be  weighed. 

It  will  be  observed  that  the  five  decimal  weights 
could  weigh  only  the  10th  parts  of  the  unit,  while  the 
five  tonal  weights  give  a  nicety  of  every  16th  part ; 
consequently  the  tonal  system  has  in  that  case  60  per 
cent,  advantage  of  the  decimal  system,  and  moreover 
the  tonal  weights  give  the  natural  and  desired  fractions, 
quarters,  eighths  and  sixteenths,  which  is  not  the  case 
with  the  decimal  weights. 

For  the  natural  fractions  it  will  require  three  more 
parts  to  the  decimal  weights,  namely  |,  ^  and  ^,  or 
expressed  by  decimals  it  will  be  0*5,  0"25  and  0125, 
by  which  the  sixteenth  parts  can  be  weighed,  but  it 
will  be  a  complicated  expression,  for  instance,  6  parts 
will  be  expressed  by  0*375  and  7=0*4375,  which  can 
never  be  clearly  comprehended,  because  the  mind  must 
be  carried  away  to  several  thousands  for  only  one 
fio-ure. 


84 

The  decimal  system  can  never  avoid  the  expression 
of  the  tonal  or  natural  fractions,  because  they  are  of 
daily  occurrence  in  practice,  while  the  tonal  system  is 
complete  in  itself  for  all  uses  without  exception,  and 
needs  no  reference  to,  but  will  do  best  without,  the 
decimal  system. 

If  three  more  parts  are  added  to  the  five  tonal 
weights,  namely,  0*2,  0-4  and  0'8,  it  can  weigh  to  a 
nicety  of  every  128  parts  of  the  unit,  the  expression 
will  have  one  decimal  (called  a  tonal)  by  which  the 
true  weight  is  clearly  impressed  on  the  mind. 

If  you  examine  all  the  papers  that  have  been  written 
on  the  subject  in  question,  including  your  own,  and 
collect  all  the  advantages  and  disadvantages  of  all  dif- 
ferent systems  in  your  memorandum,  then  examine 
well  the  tonal  system,  and  you  will  find  that  all  your 
collected  advantages  are  contained  in  the  tonal  system, 
and  all  the  difficulties  and  disadvantages  are  overcome. 
Your  most  humble  and 

Most  obedient  servant, 

John  W.  Nystrom. 


On  my  visit  in  London  I  had  the  pleasure  of  meet- 
ing James  Yates,  Esq.,  M.  A.,  F.  E,.  S.,  Vice  President, 
and  Professor  Leone  Levi,  F.  S.  A.  F.  S.  S.,  resident 
Secretary  of  the  International  Association. 

Mr.  James  Yates  was  so  kind  as  to  invite  me  to  his 
house  to  see  the  great  variety  of  French  metres  spoken 
of  in  the  preceding  letters,  which  was  indeed  a  fine  col- 
lection. The  best  form  of  the  metre  in  the  collection, 
and  the  one  best  suited  to  the  artizan  I  believe  is  the 
four  folded  one.     Among   the  ten  folded  metres  was 


85 

found  what  I  have  before  remarked,  none  of  the  same 
length,  but  they  differed  up  to  IJ  millimeters.  Other 
forms,  parts  and  divisions  of  the  metre  did  not  however 
alter  my  views,  but  rather  strengthened  my  opinion 
herein  given  on  the  subject. 

John  W.  Nystrom. 
London,  September,  1860. 


,  Esq.,  President  of  the 

Society,  Philadelplda. 

Sir  : — I  have  left  in  the  care  of  Professor  X.  a 
manuscript  on  a  new  system  of  Arithmetic,  Weights, 
Measures  and  Coins,  intended  to  be  submitted  to  your 
consideration  for  publication. 

All  the  engravings  and  types  for  the  new  figures  are 
ready  for  the  press.  In  it  you  will  find  some  corres- 
pondence with  the  Decimal  Association  in  London, 
which  is  believed  worthy  of  publication  for  the  argu- 
ment on  the  French  metre. 

Yours,  most  respectfully, 

John  W.  Nystrom, 
1216  Chestnut  St.,  Ph'da. 
Philadelphia,  Se2'>.  25,  1861. 


Philadelphia,  Oct.  19,  1861. 
Mr.  Nystrom  : 

Dear  Sir  : — I  regret  to  announce  that  the  report  of 
the  Committee  on  your  essay  that  it  recommend  that 
the  essay  be  not  published,  was  adopted  by  the  Society 
at  its  meeting,  last  evening;  and  the  MSS.  was  ordered 


86 

to  be  deposited  in  the  archives  of  the  Society,  subject 
to  your  order. 

Professor  A.  and  Professor  B.  afterwards  discussed 
your  tonal  system,  and  Dr.  C.  the  octonal  system  of 
Mr.  Taylor  of  this  city,  whose  pamphlet  was  laid  on 
our  table,  and  seems  not  to  have  been  noticed  by  you.* 
It  was  suggested  that  it  would  be  agreeable  to  publish 
some  abstract  account  of  your  system  in  the  running 
minutes  of  the  proceedings  of  the  meeting. 

Very  respectfully, 

X.,  Secretary. 


Philadelphia,  Oct.  11,  1861. 
Prof.  X.,  Secreiary^  c&c,  &c. 

Dear  Sir: — I  herewith  return  the  MSS.  of  Mr. 
Nystrom,  to  be  examined  by  the  other  members  of 
the  Committee. 

I  believe  no  other  report  will  be  necessary  than 
simply  to  recommend  for  publication,  or  the  contrary. 
Although  Mr.  N.'s  papers  have  failed  to  convince  me 
of  the  great  gain  by  substituting  the  sexta  decimal 
("  tonal")  basis  of  notation,  for  the  decimal,  yet  it  is 
interesting  and  instructive,  to  have  such  a  system  fully 
worked  out,  and  placed  before  us  in  all  its  bearings ; 
that  is,  when  it  can  be  done  by  a  philosophic  and  com- 
petent mind,  as  is  manifested  in  the  case  before  us.  I 
would,  therefore,  be  in  favor  of  publication,  at  least  as 
far  as  page  60,  which  concludes  the  main  recital.  The 
remainder,  which  is  of  equal  bulk,  is  a  correspondence 
between  Mr.  N.  and  the  officers  of  the  International 
Decimal  Association,  at  London.  To  my  own  appre- 
hension, there  is  some  defect  of  force  and  perspicuity 

*  I  was  not  there. — N. 


87 

in  their  criticisms,  affording  Mr.  N.  the  opportunity  of 
making  pretty  sharp  replies.  All  this,  while  it  tlirows 
light  on  the  subject,  and  is  spicy  enough  to  aid  in  the 
digestion,  may  be  considered  as  somewhat  of  a  repeti- 
tion. 

I  have  pencilled  down  a  few  random  comments,  and 
have  had  them  copied  on  another  sheet;  and  if  you 
please,  would  like  them,  with  this  note,  to  be  handed 
to  Profs.  A.  and  B.,  along  with  Mr.  N.'s  book.  I 
conclude  by  proposing  that  the  Committee,  meet  at 
the  hall  on  the  evening  of  the  next  meeting  of  the 
Society,  18th  inst.,  at  a  quarter  before  8  o'clock,  to 

determine  their  report. 

Very  truly,  yours, 

D. 


Hashj  comments   on  Mr.  Ntstrom's   new  hasis  of 
Arithmetical  Notation, 

(Page  8,  et  passim.)     {First  comment.) 

The  term  "binary  division"  suggests  the  neces- 
sity for  coining  a  new  word.  Binary  refers  to  a  douh- 
ling,  not  a  halving  process.  Demidial  or  dimidiary, 
(from  dimidium  half)  would  express  the  very  idea ;  but 
as  yet  there  is  no  such  word ;  nor  any  that  expresses 
the  idea.  Inasmuch  as  Mr.  N.  finds  it  necessary  to 
make  many  new  words,  this  one  is  respectfully  offered. 

(Page  21.)     {Second  comment.) 

If  this  new  system  would  afford  a  relief  from  endless 
fractions,  it  would  be  a  triumph  over  the  decimal  system ; 
but  it  does  not.  While  a  sixth  part  is  represented  in 
the  decimal  system  by  -16666  .  .  .  forever,  it  stands  in 
the  tonal  system,  -29999  .  .  .  forever.  It  works  well  for 
halves,  eighths,  sixteenths;  but  does  not  work  at  all 


88 

for  thirds,  fifths,  sixths,  and  so  on.    Yet  these  divisions 
are  continually  occurring  in  practice. 

(Page  40.)     {Third  comment.) 

Prices  are  of  every  imaginable  figure.  A  car  ride  is 
five  cents.  An  exchange  ticket  seven  cents :  a  pound 
of  sugar,  9,  10  or  11  cents.  Mr.  N.'s  system  would  be 
so  much  bothered  by  these,  that  he  would  probably 
insist  that  prices  should  be  such  as  to  make  the  work- 
ing easy.  He  is  quite  in  error  about  the  dollar  holding 
a  medium  place  among  the  monetary  units  of  the 
world,  and  therefore  having  "  a  claim  to  be  chosen  as 
a  standard."  In  calling  the  French  franc  the  smallest 
unit,  he  forgets  the  piastre  of  Turkey,  the  rial  of 
Spain,  the  drachm  of  Greece,  and  some  others.  Nor  is 
the  £  sterling  the  largest  unit;  there  is  the  milreis  of 
Portugal,  and  of  Brazil.  These  errors,  however,  are 
not  material  to  the  merits  of  the  scheme. 

(Page  57  to  60.)     {Fourth  comment.) 

The  account  of  the  Russian  stchoty  or  counting 
machine  is  interesting.  It  is  essentially  the  abacus  of 
ancient  Rome,  and  of  modern  China  and  Japan,  the 
apparatus  of  a  people  very  low  in  the  scale  of  mathe- 
matical science.  Yet  Mr.  N.  would  have  it  brought  in 
our  schools  and  counting  houses,  to  help  the  new  tonal 
system,  and  "  turn  the  mind  from  the  old  basis."  It 
would  surely  be  a  retrogade  to  put  away  the  slate  and 
pencil  for  this  machine. 

(Page  69.)     {Fifth  comment.') 

If  Mr.  N.  had  observed  the  practice  of  our  market 
people,  he  would  have  found  his  argument  against  the 
decimal  system  materially  weakened.  An  article  costs 
38  cents ;  the  buyer  hands  out  a  dollar ;  the  seller  in 
making  change,  is  sure  to  act  thus : — first  lays  down  2 


89 

cents,  to  bring  his  mind  to  40  ;  and  then  easily  makes 
up  the  remaining  60  with  a  half  dollar  and  a  dime.  So 
that  he  first  steers  for  the  nearest  ten  to  rest  upon,  and 
from  that  completes  the  operation.  He  never  thinks 
of  mentally  subtracting  38  from  100,  unless  he  be  an 
old  accountant,  or  schoolmaster. 

It  may  be  observed  that  Mr.  Alfred  B.  Taylor  of 
this  city,  constructed  an  ingenious  system  on  the 
octonal  basis.  Mr.  Pitman,  the  celebrated  phono- 
grapher,  urged  a  duodecimal  reform  ;  and  Dr.  Patterson 
used  to  mourn  that  our  arithmetic  was  not  based  upon 
12  instead  of  10. 


PkUadelphia.)  1216  Chestnut  street^  Oct.  23,  1861. 

Pkofessor  X.,  Secretary  &c.,  dec. 

Dear  Sir: — Your  favor  of  the  19th  inst.  is  at  hand, 

I  am  sorry  to  hear  the Society  did  not  deem  my 

manuscript  worthy  of  publication.  It  is  true  I  have  not 
noticed  Mr.  Taylor's  octonal  system,  as  my  tonal  system 
was  written  in  Russia  long  before  Mr.  Taylor's  octonal 
system  was  published  in  America,  and  even  if  I  had 
seen  it,  it  would  not  have  altered  one  sentiment  in  my 

manuscript.  The Society  "  suggested  that  it  would 

be  agreeable  to  publish  some  abstracts,"  which  I  suppose 
from  Mr.  D.'s  letter  to  you  dated  Oct.  11th,  would  be 
to  omit  my  correspondence  with  the  International 
Decimal  Association  in  London.  "When  my  Tonal 
system  is  published,  I  shall  omit  nothing  of  the  manu- 
script, even  my  correspondence  with  and  remarks  made 
by  the  Society  will  be  published,  as  I  desire  to 

have  the  subject  thoroughly  ventilated. 

7 


90 

It  may  be  found  that  there  ''are  some  defects  of 
force  and  perspicuity  in"  the  comments  made  by  Mr. 
D.  which  affords  me  a  second  "  opportunity  of  making 
"  a  pretty  sharp  reply." 

The  first  comment  is  "The  term  binary  division  sug- 
•'  gests  the  necessity  of  coining  a  new  word."  I  have 
only  to  refer  to  page  61,  where  the  International  De- 
cimal Association  in  London,  uses  the  same  expression. 
"  Binary"  can  be  applied  to  halving  as  well  as  doubling, 
the  difference  is  only  to  go  up  or  down  the  steps. 
The  word  "  Binary  Divisions"  is  freely  used  in  Mr. 
Taylor's  report  on  the  octonal  system.  A  binary  com- 
pound, say  chloride  of  sodium,  Na,  CI,  contains  half  of 
each  substance,  which  is  an  example  of  a  binay  halving 
process. 

Second  comment.     "  If  this  new  system  would  afford 
"  a  relief  from  endless  fractions^  it  would  be  a  triumph 
"  over  the  decimal  system,  but  it  does  not."     It  would 
indeed  be  a  triumph !  but  Mr.  D.  will  never  be  satisfied 
on  that  point,  for  let  us  even  propose  one  system  of 
arithmetic  for  each  fraction,   or  attempt  to   invent  a 
system  of  arithmetic  that  would  have  no  prime  num- 
bers, he  will  still  be  disappointed.     Can  Mr.  D.  describe 
a  circle  through  these  four  points  ..'%  and  it  will  be  a 
triumph  in  geometry.     "  While  a  sixth  part  is  repre- 
"sented  in  the  decimal  system  by  0"16666,  forever,  it 
"  stands  in  the  tonal  system  029999,  forever.    It  works 
"  well  for  halves,  eighths,  sixteenths ;  but  does  not  work 
"  at  all  for  thirds,  fifths,  sixths,  and   so  on,  yet  these 
"  divisions  are  continually  occurring  in  practice."    Fifths 
are  generally  employed  for  the  necessity  of  accommo- 
dating the  decimal  system  which  imposes  so  much  incon- 
venience upon  us.     Sixths  and  12ths  are  often  used  in 


91 


practice  as  an  improvement  on,  or  to  avoid  Sths  and 
lOths;  with  the  exception  for  the  circle,  6ths  and  12ths 
are  of  little  importance  compared  with  the  binary  frac- 
tions. In  the  following  table  are  set  down  the  fractions 
in  question,  with  one,  two,  and  three,  decimals  with  their 
errors,  in  the  Tonal,  Decimal,  and  Octonal  systems. 


Systems. 

One  Decimal. 

Two  Decimals. 

Three  Decimals. 

Fraction. 

Error. 

Fraction. 

Error. 

Fraction. 

Error. 

Tonal, 

Decimal, 

Octonal, 

Tonal, 

Decimal, 

Octonal, 

J  =0-1 
1=0-1 
>=0-5 
1  =  0-3 
J  =  0-2 

0-0416 
0-0666 
0-0416 
0-0208 
0-0333 
0-0830 

i  =  0-09 
1  =  0-16 
J-0-12 
i-0-55 
1-0-33 
i-0-25 

0-0026 
0-0066 
0-0104 
0-0013 
0-0033 
0-0280 

-  =0-299  0-00016 
6 

1=0-166  0-00066 
1    0-125  0-00065 
^=0-555  0-00008 
i  =  0-333  0-00033 
1=0-2520-00130 

It  will  be  seen  in  this  table  that  the  fraction  l  ex- 
pressed by  one  decimal  has  by  the  tonal  system  60  per 
cent  advantage  in  the  correctness  over  the  decimal  sys- 
tem. With  two  decimals  250,  and  with  three  decimals 
410  per  cent,  advantage  in  the  correctness.  Still  Mr. 
D.  says  these  fractions  "does  not  work  at  all."  For 
thirds  the  favor  is  still  greater  for  the  tonal  system. 

Third  Comment.  "Prices  are  of  every  imaginable 
"  figure.  A  car  ride  is  five  cents,  an  extra  ticket  7 
"  cents."  In  my  manuscript  I  speak  about  omnibus 
prices,  as  they  were  when  I  left  America  for  Europe  in 
the  spring  of  1856,  which  was  written  in  Russia  in  the 
year  1859,  when  I  knew  nothing  about  the  street  rail- 
road arrangement.  Six  cents  or  rather  6J  is  a  very 
general  price  for  articles,  as  being  -j^th  part  of  a  dollar. 
After  my  return  from  Europe,  I  made  the  following 
observations  on  car  ride  prices : 


92 


Frcmhford  and  SoufhivarJc  Passenger  JR.  li.  Co. 

FARES. 

Southwark  to  Front  and  York  streets,  5  Cts. 

Frankford,  10 

Germantown  road  to  Frankford,  7 

Southwark  to  Episcopal  Hospital,  7 

Berks  street  to  Harrowgate,  5 

Frankford  to  Hart  lane,  5 

For  Children  under  12  years,  3 

By  examining  this  price  table  we  find  that  all  the 
prices  are  not  only  odd  but  of  prime  numbers  3,  5  and 
7.  The  base  price  for  a  car  ride  is  5  cents,  and  for 
long  distances  double  price,  10  cents.  For  interme- 
diate distances  such  as  from  Germantown  to  Frank- 
ford, and  from  Southwark  to  the  Episcopal  Hospital, 
is  charged  7  cents,  showing  an  attempt  to  charge  a 
price  half  way  between  5  and  10,  which  should  be  7| 
cents,  bat  our  coins  as  well  as  our  decimal  base  does 
not  permit  such  division,  for  which  we  must  be  con- 
tented with  the  prime  number  7.  It  is  a  general 
custom  over  the  world,  to  charge  half  price  for  children, 
which  in  this  case  should  be  2|  cents,  but  as  we  have 
no  such  coins,  it  is  made  up  to  3  cents. 

Let  us  now  see  what  the  to?ial  prices  would  be.  By 
the  tonal  system  it  is  very  likely  that  the  base  price 
for  a  car  ride  would  be  1  shilling,  (6^  cents,)  but  sup- 
pose even  this  to  be  too  high,  and  the  exact  value  of 
5  cents  is  required,  which  would  be  (9  tonal  cents, 
the  price  table  would  be  as  follows : 


Southwark  to  York  Street, 

Frankford, 
Germantown  rd,  to  Frankf 'd, 
Berk  street  to  Harrowgate, 
Children  half  price, 


Decimal. 

Tonal 

Prices. 

5  Cts. 

'19  cts. 

1  s. 

10 

1-8  s. 

2 

i    7 

1-2 

1-8 

5 

C?  cts. 

1 

3 

6 

8  cts. 

93 

The  tonal  price  in  both  cases  are  all  of  even  and 
easy  countable  numbers,  and  divides  the  prices  as  de- 
sired in  practice. 

The  Sanford's  Opera  bill  says : 

Admittance,         -         -         -     25  cents. 
Half  price  for  children,  -     13     " 

Pennsylvania  Railroad  trains  leave  Philadelphia  at : 


Mail  train. 
Fast  line. 
Through  express, 
Harrisburg  train, 

Dpcimal. 

Tonnl. 

8  A.  M. 
11-30  A.  M. 
10-30  P.  M. 

2.30  P.  M. 

5-8  T. 

6 
9-9 

In  this  time  table  there  is  no  confusion  of  A.  IM.  and 
P.  M.  in  the  tonal  column,  but  the  correct  time  is  ex- 
pressed by  the  fewest  possible  numbers,  clear  to  the 
mind  at  the  first  glance.     T  means  the  hour  mark. 

The  original  meaning  of  A.  M.  and  P.  M.  is  not 
generally  known  further  than  that  it  means  forenoon 
and  afternoon,  but  even  that  is  sometimes  confused. 

Two  coal-miners,  Jack  and  Harry,  arrived  at  a  rail- 
way station  in  England,  and  examined  the  time  table, 
when  the  following  conversation  took  place : 

Jack.  I  say  Harry,  what  does  P.  M.  mean  % 

Harry.  Penny  a  mile,  to  be  sure. 

Jack.  What  does  A.  M.  mean,  then  \ 

Harry.  Oh,  that  must  be  a  A  penny  a  mile. 

Jach.  Then  we  will  go  by  the  A.  M.  line. 

Jack  and  Harry  were  not  acquainted  with  ante  meri- 
diem and  jpost  meridiem. 

In  the  crowded  railway  guides  it  is  often  difficult  to 
find  out  whether  a  noted  time  means  in  the  forenoon 
or  afternon,  and  it  is  often  necessary  in  tables  to  leave 
a  separate  column  for  A.  M.  and  P.  M. 


94 

Rev.  E.  Barnham  preaches  in  the  Concert  Hall, 
Philadelphia,  every  Sunday  at: 

10 J  A.  M.,  3  P.  M.,  and  7 J  evening. 

Tonal  time,  7  9  8  T. 

Our  present  system  employs  tivelve  characters  and 
one  word,  where  the  tonal  system  uses  only  three 
characters  and  the  hour  mark.  The  Pev.  gentlemen 
seems  disposed  to  divide  his  time  as  in  the  tonal  system. 

Third  Commemt.  "  Mr.  Nystrom's  system  would  be  so 
"  much  bothered  by  those  that  he  would  probably  insist 
"  that  prices  should  be  such  as  to  make  the  working  easy." 

Every  shop  keeper  attempts  to  arrange  his  prices 
into  easy  countable  figures,  but  the  inconvenience  of 
the  decimal  system  is  so  great  that  it  is  difficult  or 
rather  impossible  to  satisfactorily  attain  that  object,  as 
is  readily  seen  in  store  windows  by  prices  marked  on 
articles  37|  cents,  81^  cents,  &c.,  &c.,  an  attempt  to 
approach  the  tonal  system.  If  Mr.  D.  will  take  the 
trouble  to  examine  the  shop  practice,  he  may  discover 
that  prices  are  already  arranged  to  suit  the  tonal 
system,  in  spite  of  the  bothered  decimal  coins. 

Fourth  Comment,  "  He  is  quite  in  error  about  the 
"  dollar  holding  a  medium  place  among  the  monetary 
"  units  of  the  world."  How  does  Mr.  D.  know  that  the 
dollar  is  not  holding  a  medium  place,  among  monetary 
units '?  Did  he  try  it  %  And  if  so,  why  not  favor  us 
with  his  result?  Is  it  more  or  less]  The  following 
table  contains  the  present  monetary  unit  in  most  parts 
of  the  world. 

The  table  can  however  be  varied  by  judgment  of 
different  units  existing  in  different  countries,  that  any 
one   who    feels    disposed    to    make    comments   on    it 


95 


has  an  extensive  field  to  operate  upon.  When  I  first 
worked  out  this  medium  monetary  unit  the  result 
came  much  nearer  the  dollar  than  in  this  table. 

Monetary  units  in  most  parts  of  tlie  world. 


South  American  &  > 

Mexican  dollar,   S 

Chili    and    New    > 

Granada  dollar,    S 
China  dollar, 
Austria,  Bohemia  > 

&  Bavaria,  Florin,  S 
Denmark,  Specidaler, 
France,    Belgium, 

Switzerland  and 

Italy,  Franc, 
Great  Britain,  £ 

Sterling, 
India,  Rupies, 
Hamburg,  Mark, 
Hanover,  llialer, 
Holland,  Florin, 
Prussian,  Thaler, 
Portugal,  Millrea, 
Pome,  Scudo, 
Pussian,  Ruble, 
Saxony,  Thaler, 


$.    cts. 

1-05 

0-96 

1-43 

0-50 
0-96 

0-18 

4-86 

0-46 
0-30 
1-10 
0-41 
0-72 
M2 
1-04 
0-77 
0-63 


Spanish,  Piaster, 
Sweden,  Riksdaler, 
Turkey,  Piastre, 
Wirtemberg,  Thaler, 

Total, 

Divided  by  20  will 
cents  as  a  medium  unit. 

The  main  money  of 
Europe    is    Pound 


$.   cts. 

106 
0-27 
004 
1-00 


18-86 
oe  94 


Sterling, 
Florin  of  Germany, 
Thaler  of  Germany, 
Puble  of  Pussia, 
Franc  of  France, 
Piksdaler  of  Sweden, 


7-23  :  6  =  1-20 
0-94 

2  I  2-14 


4-86 
0-45 
0-70 
0-77 
0-18 
0-27 


7-23 


1-07  dels. 

Which  can  be  considered 
a  medium  monetary  unit  of 
the  world. 


Fifth  Comment.  "  In  calling  the  franc  the  smallest 
"  unit  he  forgets  the  piaster  of  Turkey,  the  real  of  Spain, 
"  the  drachm  of  Greece,  and  some  others.  Nor  is  the 
"  £   sterling   the    largest    unit ;    there    is   medries   of 


96 

"Portugal,  (§1  12)  and  of  Brazil."  I  beg  to  assure  Mr 
D.  that  tlie  units  referred  to  are  not  forgotten. 

If  we  go  to  those  extremes  it  is  difficult  to  know 
where  to  begin  or  end.  We  may  call  the  American 
eagle  10  dollars  a  unit,  the  Napoleon  20  franc.  Impe- 
rial 100  fr,  doubloon  of  Spain,  Central  and  South 
America,  about  15  dollars.  Russian  imperial  5*15 
Rubles.  Dobras  of  Brazil  34  dollars,  and  other  monetay 
units  rapj^ins:  from  a  fraction  of  a  cent  towards 
50  dollars.  If  we  go  further  to  the  corners  of  the 
land  among  the  Esquimaux's  or  the  Calmucks  we 
may  find  a  liide  of  any  animals  or  a  heap  of  hay  to  be 
a  unit  for  trading.  The  £  sterling  and  franc  are  the 
extreme  units  known  and  handled  over  the  whole 
world,  in  preference  of  which  the  outside  units  of 
Greece  and  of  the  Esquimaux  could  not  be  admitted 
in  a  general  statement.  The  French  Franc  is  used  in 
Belgium,  Switzerland,  Italy  and  Algiers,  by  a  popu- 
lation of  about  70  million ;  the  £  sterling  is  used  by  a 
population  of  I  suppose  50  millions. 

The  franc  and  £  sterling  put  together,  I  believe 
would  exceed  all  the  rest  of  the  money  in  the  world. 
Mr.  D.  says  "  These  errors  however  are  not  mate- 
rial to  the  merits  of  the  scheme."  I  have  not  been 
able  to  find  a  single  expression  in  Mr.  D.'s  comments 
that  have  any  bearing  whatever  on  merit  or  folly 
of  the  tonal  system. 

Sixth  Comment.  "  Yet  Mr.  Nystrom  would  have  it 
"  (the  counting  machine)  introduced  into  schools  and 
"  counting  houses  to  help  the  new  tonal  system  and 
"  turn  the  mind  from  the  old  basis.  It  would  surely 
"  be  a  retrograde  to  put  away  the  slate  and  pencil  for 
"  this  machine."     I  would  like  very  much  to  see  Mr. 


97 

D,  with  a  slate  and  pencil  alongside  a  Russian  with 
a  tshoty,  I  would  give  the  example  for  calculation, 
and  Mr.  D.  would  soon  find  the  utility  of  the 
Russian  tshoty,  and  the  retrograde  of  his  slate  and 
pencil.  When  the  tonal  system  is  well- acquired,  the 
counting  machine  would  be  found  superfluous,  but  for 
the  first  acquirement  the  slate  and  pencil  would  not 
answer  the  same  purpose  as  the  counting  machine. 

Seventh  Comweni.  "  If  Mr.  Nystrom  had  observed 
"  the  practice  of  our  market  people,  he  would  have 
"  found  his  argument  against  the  decimal  system 
"  materially  weakened."     After  having  read  Mr.  D.'s 

hasty  comments  in  the  Archives  of  the Society, 

I  proceeded  up  Chestnut  street,  when  opposite  the 
Masonic  Hall  I  observed  at  the  northwest  corner 
of  Eighth  and  Chestnut,  at  the  doorway  of  Sharp- 
less'  store,  a  pile  of  dry-goods  upon  which  was  a 
paper  sign  marked  with  figures  as  big  as  my  hat, 
"Extra  quality  87 J  cts."  (per  yd.)  Arriving  at  the 
southeast  corner  of  Eighth  and  Chestnut  streets,  I  saw 
at  the  Eighth  street  doorway  of  the  same  store,  two 
piles  of  dry-goods  marked  one  37|  cts.  and  the  other 
62|  cts.  (per  yard).  Looking  up  Eighth  street  I  saw 
at  the  northeast  corner  of  Eighth  and  Zane  streets, 
in  the  doorway  of  a  store,  a  crinoline  in  the  inside  of 
which  was  a  paper  sign  marked  37|  cts.  I  walked 
up  to  this  store  where  1  found  in  the  window  fifteen 
articles  marked  with  the  following  prices,  ^\  cts. 
12J  cts.  18f  cts.  25  cts.  31J  cts.  371  cts.  &c.,  &c.,  all 
arranged  to  accommodate  the  easy  counting  as  in  the 
tonal  system. 

Returned  and  went  into  Mitchell's  restaurant,  808 
Chestnut   street,  where  I  took   a   cup    of  coffee   and 


98 


cakes;  was  handed  an  ivory  ticket  upon  which  was 
engraved  the  number  19,  indicating  the  price  of  my 
refreshment.  At  the  counter  I  inquired  why  they 
charged  the  odd  number  19  cts.  and  was  answered 
"  It  ought  to-be  18|  cents,  but  as  we  have  no  such 
"  coins  we  make  it  even  to  19."  I  then  asked,  would 
not  20  be  a  more  even  number]  And  was  answered, 
"  20  is  not  even  in  a  dollar."  Here  you  will  find  that 
the  prime  number  19  is  called  even,  and  the  even 
number  20  is  considered  to  be  odd. 

It  is  literally  true,  that  20  is  odd  in  100,  and  that 
odd  numbers  as  5,  25  and  75,  are  even  in  100. 

Upon  further  inquiry  of  their  prices,  I  was  shown 
to  the  other  side  in  the  store,  to  a  box  of  about  13 
inches  square,  divided  into  36  compartments,  each  con- 
taining ivory  tickets  marked  with  the  following  prices 

MitchelVs  Price  Ticket  Box. 


6ict. 

13 

19 

25 

31 

38 

44 

50 

56 

63 

69 

75 

81 

88 

94 

100 

106 

1 
112  ' 

118 

125 

131 

137 

144 

150   : 

: 
j 

162 

168 

175 

181 

200 

275 

281 

287 

300 

306 

312 

1 

318  1 

1 

99 

Here  you  will  find  that  it  is  attempted  to  arrange 
the  prices  to  suit  the  dollar  divided  into  16  parts,  as 
in  the  tonal  system.  You  will  observe  that  most  of 
the  prices  are  of  odd  and  prime  numbers,  and  to  make 
them  perfectly  correct,  most  of  them  ought  to  be  ac- 
companied with  fractions.  It  is  evident  that  those 
prices  are  considered  easy  to  the  mind  in  the  market, 
and  how  much  better  would  it  not  be,  if  our  arith- 
metic was  based  on  the  same  principle ;  every  coin 
proposed  in  my  tonal  system  agrees  correctly  with 
those  prices. 

Taylor's  saloon  in  New  York,  and  a  great  many 
other  establishments,  have  similar  arrangements  of 
prices. 

I  left  the  restaurant,  walked  up  Chestnut  street, 
stopped  at  the  store  of  Le  Boutillier  Brothers,  No. 
912,  where  I  found  prices  marked  in  the  window  62| 
cents,  87|  cents,  75  cents,  &c.,  &c. 

At  Besson  &  Son's  mourning  store,  918  Chestnut 
street,  I  found  in  the  window  marked  the  following 
prices. 

De  Laines,  12J  cents.  Reps  Anglais,      37J  cents. 

Cravellas,  25    cents.  Mousselin,  6^  cents. 

De  Laines,  18,|  cents.  Other  articles,     62 J  cents. 

Grandrill,  31 J  cents.  One  article,         44    cents. 

No  price  had  any  indication  of  decimal  division. 
Went  home  to  my  house  1216  Chestnut  street,  where 
I  pay  for  my  washing  62J  cents  per  dozen.  About 
two  weeks  ago,  I  was  charged  by  a  shoemaker  37 J 
cents  for  mending  a  pair  of  boots. 

AVill  Mr.  D.  yet  think  that  my  observation  of 
market  practice  would  materially  weaken  my  argument 


100 


against  the  decimal  system'?  and  in  case  he  suppose 
that  this  is  my  first  attention  to  market  practice,  I  beg 
to  remind  him  that  in  my  manuscript  it  is  plainly  stated 
that  it  is  my  observation  of  the  inconvenience  of  the 
decimal  system  and  arithmetic  in  the  shop  and  market 
that  has  led  me  to  propose  the  tonal  system.  In  Mr. 
D.'s  argument  on  the  38  cents,  he  still  brings  the 
mind  to  the  high  numbers  of  40,  60  and  100,  where 
the  tonal  system  would  bring  it  only  to  the  base  10. 
The  price  38  cents  would  be  6  shillings  tonal. 

.  The   following    table    contains    the   most    common 
market  price. 


Market  Prices  in  Cents. 

Tonal  Shillings  or  IGths  of  a 
Dollar. 

Nearest  Cents  as  Mitchell's 
Ticket-. 

6i 

1 

6 

12i 

2 

12  or  13 

18.1 

3 

19 

25 

4 

25 

,       3ii 

5 

31 

37J 

6 

37  or  38 

43i 

7 

44 

50 

8 

50 

56J 

9 

56 

62i 

10 

62  or  63 

681 

11 

69 

75 

12 

75 

81i 

13 

81 

m 

14 

87  or  88 

93| 

15 

94 

100 

16 

100 

Can  Mr.  D.  discover  any  utility  in  the  centre 
column  of  this  table,  compared  with  the  two  outside 
ones'? 


101 

Lastly,  Mr.  D,  says :  "  It  may  be  observed  that 
"  Mr.  Alfred  Taylor  of  this  city,  lately  constructed  an 
"  ingenious  system  on  the  octonal  basis.  Mr.  Pitman, 
"  the  celebrated  phonographer  used  a  duodecimal  reform ; 
"  Dr.  Patterson  used  to  mourn  that  our  arithmetic  was 
"not  based  upon  12,  instead  of  10."  It  seems  from 
these  statements,  that  Mr.  D.  has  no  preference  to 
any  one  of  the  three  basis,  8,  12  and  16,  that  if  I  had 
proposed  14  as  a  base,  he  may  have  given  it  the  same 

consideration ;  and  if  such  is  the  case  with  the 

Society,  I  do  not  wonder  at  all,  that  the  tonal  system 
was  rejected  for  publication. 

There  is  nothing  new  in  merely  proposing  a  better  base 
for  our  arithmetic ;  that  I  believe  has  been  done  since 
the  time  of  Charles  XII.,  of  Sweden,  by  hundreds,  and 
been  thought  of  by  thousands  ;  for  any  self  thinker  with 
good  reason  of  mind,  sees  plainly  the  foUy  of  our  deci- 
mal arithmetic.  I  am  surprised  to  find  so  many  of  the 
first  leaders  of  the  scientific  world  to  be  so  short 
sighted,  as  not  to  see  the  inconveniences,  but  propa- 
gates a  system  so  unnatural  in  all  its  bearings. 

About  a  year  ago  there  was  an  Italian  in  London, 
who  proposed  the  duodecimal  system,  but  in  no  case  have 
I  found  any  of  such  systems  worked  out  with  examples 
into  a  practical  shape.  Most  of  the  propositions  have 
been  made  by  mere  scientific  men,  who  have  given 
excellent  accounts  of  the  history  of  arithmetic,  and 
finished  by  merely  proposing  a  better  base.  I  believe 
myself  to  have  commenced  and  continued  from  where 
they  ended,  and  I  suppose  you  to  know  what  they  have 
said. 

Purely  scientific  men  are  not  the  proper  persons  to 
handle  this  practical  subject,  for  the  decimal  arithmetic 


102 

is  so  clear  to. them,  that  they  manage  the  figures  and  come 
to  their  results  as  easy  as  a  musician  who  plays  the  crank 
organ.  Their  lack  of  direct  application  of  their  science 
to  practice,  screens  away  the  real  inconvenience  of  our 
decimal  arithmetic,  which  is  readily  proved  by  feeble 
remarks  frequently  made  by  such  men.  Many  of  them 
confine  themselves  more  to  style  of  language  than  to 
the  substance  of  the  subject.  It  is  not  sufficient  merely 
to  propose  or  say  that  8,  12  or  16,  would  be  better  as 
a  base,  but  in  order  to  make  a  clear  and  correct  im- 
pression of  its  utility,  it  is  necessary  to  enter  into  details 
with  examples,  that  any  one  may  be  able  to  estimate 
its  advantages  without  taxing  his  own  mind.  Still  the 
nature  of  the  subject  is  such,  as  to  be  apt  to  be  called 
curious  at  the  first  glance. 

The  octonal  system  has  two  serious  objections: 

First.  That  the  base  8  is  too  small.  Our  experi- 
ence with  the  decimal  arithmetic  is,  that  10  is  too 
small  as  a  base. 

Secondly.  As  we  progress  in  this  world,  generation 
after  generation,  we  require  larger  and  larger  numbers 
in  our  transactions.  That  which  Moses  counted  by 
thousands,  are  by  us  counted  by  millions,  and  I  ven- 
ture to  say,  that  with  the  present  decimal  arithmetic, 
there  are  very  few  who  have  a  clear  conception  of  the 
immense  number  of  one  million ;  and  the  more  com- 
plication we  have  to  lead  us  to  such  a  number,  the 
more  cloudy  it  will  be  to  the  mind.  The  immense 
numbers  necessary  in  astronomy,  expressed  by  decimal 
arithmetic  are  inconceivable,  while  the  toned  system 
gradually  leads  the  mind  towards  infinitum. 

In  the  octonal  system  we  have  two  figures  already  at 
8,  three  at  64,  and  four  at  512 ;  while  in  the  tonal 


103 

system  we  have  two  figures  first  at  16,  three  at  256, 
and  four  at  4096.  One  decimal  million  expressed  by 
odonal  arithmetic  will  be  3,641,100,  and  by  tonal 
arithmetic  94,240,  which  is  a  difference  of  two  figures. 
Also  for  decimal  fractions  the  odonal  requires  more 
figures  than  the  ional  system  for  the  same  nicety. 
The  duodecimal  has  many  advantages  over  the  decimal 
system,  particularly  in  thirds  and  sixths,  but  this  is 
overbalanced  by  the  serious  objection  of  it  not  admit- 
ting binary  division  to  infinitum.  Sixths  and  thirds 
work  much  better  in  the  tonal  than  in  the  decimal 
system,  as  seen  in  the  table,  page  91. 

Many  self-thinkers  express  their  regrets  that  the 
arithmetical  system  was  not  from  the  beginning 
founded  on  a  better  base.  Many  of  them,  I  believe, 
prefer  the  duodecimal  system,  and  some  express  their 
wish  that  man  would  have  had  six  fingers  on  each 
hand,  which  might  have  led  to  a  duodecimal  system. 
The  Sixdiopt  Family,  in  Central  America,  have  six 
fingers  on  each  hand,  and  six  toes  on  each  foot;  they 
might  have  had  accomplished  that  object.  By  this 
theory,  I  would,  of  course,  prefer  eight  fingers  on  each 
hand.  I  know  no  tribe  of  people  that  can  accommodate 
me,  but  am  satisfied  that  five  fingers  will  answer  for  the 
tonal  system.  Should  we  now  succeed  to  introduce  a 
duodecimal  system,  our  descendants  tvoidd  surely  wish 
for  the  toned  system  of  2cs,  as  we  wish  for  a  duodecimal 
system  from  our  ancestors. 

If  the  Arabic  notation  employed  in  our  present  decimal 
arithmetic  had  been  suggested  to  Moses  when  he 
wrote  the  ten  commandments  on  Mount  Sinai,  he 
w^ould  surely  have  made  similar  remarks  as  that  made 


104 

on  the  tonal  system,  that  such  curious  looking  diardciers 
could  not  be  understood  by  his  people. 

My  manuscript  on  the  ional  system  has  been  sent  to 
a  great  many  places  for  publication.  The  Franklin 
Institute  thought  it  would  have  a  very  serious  effect 
on  the  number  of  subscribers  of  their  journal !  The 
Smithsonian  Institute  would  not  publish  it,  because 
they  had  so  much  of  the  same  kind  before ! ! !  The 
U.  S.  Coast  Survey  stated  they  would  publish  it  if 
recommended  by  a  member  of  Congress.*  And  lately 
the Society  of  Philadelphia  has  rejected  it,  per- 
haps on  the  remarks  herein  replied  to,  but  Mr.  D. 
recommended  its  publication. 

I  return  herewith  the  pamphlet  on  Mr.  Taylor's 
octonal  system,  and  thank  you  very  much  for  calling 
my  attention  to  it.  It  is  indeed  an  interesting  and 
ably  written  work.  I  suppose  I  must  follow  the  track 
of  Mr.  Taylor,  and  go  to  Boston  to  get  my  tonal 
system  understood  and  appreciated. 

The  heading  of  this  letter  is  dated  October  23d, 
when  I  intended  to  write  but  a  few  lines,  but  when  I 
got  into  it,  I  could  not  well  cut  it  off  until  it  reached 
nearly  twenty-four  pages.  It  is  now  the  28th  of 
October. 

Your  humble  and  obedient  servant, 

John  AV.  Nystrom. 


*  The  idea  of  showing  to  a  member  of  Congress  this  manuscript  and 
calculations,  with  Vs,  ^s,  &c.,  among  the  figures !  he  would  surely  pro- 
nounce me  a  funny  fellow.     When  scientific  men,  as  at  the  Franklin  and 

Smithsonian  Institutes,  Society,  and  others,  cannot  appreciate  the 

subject,  what  can  we  then  expect  of  a  member  of  Congress  ? 


105 


XYSTROM'S   CALCULATOR. 


This  calculating  machine  consists  of  a  silvered  brass 
plate  of  about  nine  inches  in  diameter,  on  which  are 
fixed  two  movable  arms,  extending  from  the  centre  to 
the  periphery.  On  the  plate  are  engraved  a  number 
of  curved  lines  in  such  form  and  divisions  that  with 
their  intersection  with  the  arms,  the  most  complicated 
calculations  can  be  performed  almost  instantly. 

The  arrangement  for  trigonometrical  calculations  is 
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sine^  cosine^  tangent^  &c.,  operating  only  by  the  angle 
expressed  in  degress  and  minutes,  and  without  any 
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Engineering  have  been  computed  by  this  instrument. 


106 

Teachers  are  generally  dependent  upon  text  book 
for  examples,  when  it  is  easy  for  the  pupil,  knowing 
where  it  comes  from,  to  be  furnished  with  an  answer; 
but  with  the  calculator,  the  teacher  can  vary  the 
examples  ad  Uhitum,  and  the  answer  is  almost  instantly 
at  hand,  while  the  pupil  is  thrown  on  his  own 
resources  for  the  proper  solution,  and  his  real  acquire- 
ment is  tested. 

The  price  of  the  Calculator,  with  complete  descrip- 
tion and  examples  how  to  use  it,  $20. 

Manufactured  by  Wm.  J.  Young,  43  North  Seventh 
Street.  Sold  by  James  W.  Queen  «fe  Co.,  924  Chestnut 
Street,  Philadelphia. 


NYSTROM'S 

PUBLISHED    BY 

J.   B.    LIF»P»I]SrCOTT    &    CO. 

PHILADELPHIA. 

TRUBNEE   k   CO.,  LONDON. 

This  Pocket  Book  is  now  in  its  fifth  edition,  revised 
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v*n.t'£r*tua^>->>     >>it*.i: 


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